- cmap, École polytechnique latent liquidity models: market...

Iacopo Mastromatteo - CMAP, École Polytechnique

Latent liquidity models: An universal mechanism for price impact

Market Microstructure, confronting many viewpoints

Paris,December 8-11, 2014

J. BonartJ.-P. BouchaudJ. DonierB. Tóth

Le logotype en réserve blanche s’applique sur fonds foncés.

Le logotype bleu, noir… … ou blanc

Le logotype, dans ses versions verticale et horizontale, existe en bleu pour un usage sur fond clair.

La version en noir s’applique à titre exceptionnel, uniquement pour certains modèles d’enveloppes.

PANTONE 303CCMJN 100 - 0 - 0 - 75RVB 0 - 62 - 92

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9

[see arXiv:1412.0141]

Outline

❖ Motivation: empirical results and phenomenological formula for anomalous price impact

❖ Idea: an intuition about latent liquidity models

❖ Model: a consistent model for anomalous price impact

❖ Results: analysis, numerics and predictions

❖ Discussion: extensions and open problems

Motivation

The problem of market impact

Buy trades push prices up. Sell trades push prices down.A trivial statement

How much do buy (sell) trades move prices up (down)?For how long?

A not-so-trivial question

How to measure this? How to model it?In the following, we will study:

A hard question

I(Q) = h(pT � p0)|Qi

Relevance of market impact

❖ Theory: Relevant, because market impact is the mechanism through which prices absorb information encoded in trades, making prices efficients;

❖ Practice (I) : Market impact is a cost for traders, which they need to accurately control in order to optimize execution;

❖ Practice (II) : For regulators, market impact controls stability.

Why is this issue relevant?

square-root impact law appears to hold approximately inall cases.

The aim of the present paper is to provide a theoreticalunderpinning for such a universal impact law. We first givea general dynamical theory of market liquidity that predictsthat the average supply/demand profile is V shaped aroundthe current price. The anomalously small local liquidityinduces a breakdown of the linear response and explainsthe square-root impact law. We then study numerically astylized model of order flow based on minimal ingredients.The numerical results fully support our analytical argu-ments and allow us to get quantitative insights into variousaspects of the problem.

II. AN INTRIGUING IMPACT LAW

One should first carefully distinguish the total impact ofa given metaorder of sizeQ from other measures of impactthat have been reported in the literature. One is the imme-diate impact of an individual market order of size q, whichhas been studied by various authors and is also stronglyconcave as a function of q, i.e., q! with ! ! 0:2, or evenlnq [3]. Another easily accessible measure of impact is torelate the average price change !T in a given time intervalT to the total market order imbalanceQT in the same time

period, i.e., the sum of the signed volumes of all marketorders. This quantity is estimated using all the trades in themarket (i.e., those coming from different market partici-pants) and is clearly different from the impact of a givenmetaorder (see below). However, there seems to be quite abit of confusion in the literature and many authors undulyidentify the two quantities. If T is very short, such thatthere are only one or a few trades, one essentially observesthe concave impact of individual orders that we just men-tioned. But as T increases, and as such, the number oftrades becomes large, the relation between !T and QT

becomes more and more linear for small imbalances (see,e.g., [3], Fig. 2.5), and on time scales comparable to thoseneeded to complete a metaorder, the concavity has almostdisappeared, except in rare cases when QT=V is large—inany case, much larger than the region where Eq. (1) holds.A square-root singularity for small traded volumes is

highly nontrivial, and certainly not accounted for in Kyle’sclassical model of impact [11], which predicts a linearimpact ! / Q. A concave impact function is often thoughtof as a saturation of impact for large volumes. We believethat the emphasis should rather be placed on the anomaloushigh impact of small trades. Numerically, Eq. (1) meansthat trading 100th of the daily volume moves the price by atenth of its daily volatility, which is indeed a huge ampli-fication. Mathematically, Eq. (1) implies that the marginal

impact diverges for small volumes as Q"1=2, which meansthat the susceptibility of the market to trades of vanishingsize is formally infinite. In most systems, the response to asmall perturbation is linear, i.e., small disturbances lead tosmall effects. The breakdown of the linear response oftenimplies that the system is at, or close to, a critical point,where very special properties emerge, such as long-rangememory or scale-invariant avalanches, that accompany thisdiverging susceptibility. Of course, the mathematical di-vergence is cut off in practice—for one thing, the volumeQ of a metaorder cannot be smaller than a single lot.Empirical data will never be in the asymptotic limitQ=V ! 0, but this is irrelevant to our discussion. This is,in fact, also the case for most physical systems for whichcritical behavior is observed. The important point here isthat the proximity of a critical point can lead to stronglynonlinear effects and extreme fragility. As we will argue indetail below, and substantiate within a precise numericalmodel, we believe that markets operate in a critical regimewhere liquidity vanishes. This offers a framework to under-stand many of the anomalies in the behavior of markets,including the long-term memory in order flow and thepresence of frequent unexplained jumps in prices, thatare—or so we believe—a consequence of the chroniclack of liquidity that leads to a micro crisis. The anomaloushigh impact of small trades implied by the concave impactlaw, Eq. (1), is, in our view, another side of the same coin.Numerous interpretations have been put forth to explain

a concave impact law, and can be broadly classified into

10-5

10-4

10-3

10-2

Q/V

10-3

10-2

10-1

∆/σ

Small ticksLarge ticksδ=1/2δ=1

Impact

Jun. 2007 - Dec. 2010

FIG. 1. The impact of metaorders for Capital FundManagement proprietary trades on futures markets, in the periodfrom June 2007 to December 2010. Impact is measured here asthe average execution shortfall of a metaorder of size Q. Thedata base contains nearly 500 000 trades. We show !=" vs Q=Von a log-log scale, where " and V are the daily volatility anddaily volume measured the day the metaorder is executed. Theblue curve is for large tick sizes, and the red curve is for smalltick sizes. For large ticks, the curve can be well fit with # ¼ 0:6,while for small ticks we find # ¼ 0:5. For comparison, we alsoshow the lines of slope 0.5 (corresponding to a square-rootimpact) and 1 (corresponding to linear impact). We have re-moved a small positive intercept !=" ¼ 0:0015 for Q ¼ 0,which is probably due to a conditioning effect.

B. TOTH et al. PHYS. REV. X 1, 021006 (2011)

021006-2

Empirical results on MI

Evidence dating back to 1997 (!) shows that impact has a concave shape (roughly) independent of:

- Venue- Maturity- Historical period- Geographical area

[see Torre (1997), Almgren et al. (2003), Moro et al. (2009), Tóth et al. (2011), Gomes, Waelbroeck (2014), Bershova, Rakhlin (2013), IM et al. (2014), X. Brokmann et al. (2014), Zarinelli et al. (2015)]

[from Tóth et al (2011), impact of ≈ 500 000 trades]

Q/V

I(Q)/�

Phenomenological formula for MI

❖ Market fragility: A concave dependence implies large response to small trades (1% of daily volume moves by 10% of daily volatility)

❖ Non-additivity: When executing 2Q trades sequentially, the effect of the second group of trades is smaller

❖ Scale-independence: Volatility scales with T1/2 and volume with T. So, the law has no scale.

The shape of the impact function is well-fitted by the following parameterization:

I(Q) = Y �

✓Q

V

◆1/2

where we notice:

price change

executed volume

daily volatility

daily traded volume

Y-ratio(adimensional ∼ 1)

I(Q)

Q

V

Y

�

The origin of (mechanical) impact

Limited liquidity. If liquidity was infinite, everyone could instantly execute,and traders wouldn’t need to split their trades and execute incrementally…

Why does transient impact arise?

Why is liquidity limited?Because price exist! Due to market clearing, you don’t have any

liquidity at the current price by definition.

We want to incorporate these generic (universal) ingredients in a stylized model,in order to see how much of the phenomenon we can capture.

Particle price vs interface price

Existing orders are arranged on a price line: their interface defines the price

The actual microstructure motivates a model in whichprices are not fundamental, but emergent objects: prices evolve through orders.

Dem

and⇢

B(x,t)

Supp

ly⇢

A(x,t)

x < pt, demand levels x > pt, supply levels

Price pt

DemandSupply

Intuition for concave impactLet’s zoom in around the current price. For a non-degenerate density,

⇢A(x� pt) ⇡ ⇢B(pt � x) ⇡ LxD

eman

d⇢

B(x,t)

Supp

ly⇢

A(x,t)

x < pt, demand levels x > pt, supply levels

Price pt

DemandSupply

Q = ±Z pt±�p

pt

⇢A/B(x) = L�p

2

2

QUESTION:How much does the instant executionof a volume Q affect the price?

�p =

r2Q

L

Orders of magnitude estimation…What we stated in the last slide is valid in a specific setting:

renewal time of the demand/supplycurve

duration of the order

otherwise you expect the impact to cross-over to a trivial, linear region.

T ⇠ (hours)

⌫�1OB ⇠

The correct interpretation of the demand and supply curve cannot be the order book!

(seconds)

Need for a notion of latent liquidity

(hours)

⌫�1 � T

⌫�1 ⇠

Latent order book

Fast, small liquiditySlow, large liquidity

Empirical: daily traded volume vs instantly available volume

Theoretical: no incentive in giving away private information

Latent vs Real

/ ⇠ 102

Evidence:

(

[J-P Bouchaud, J D Farmer, F Lillo,Handbook of Financial Markets:

Dynamics and Evolution (2009)]

Bid

⇢

B(x,t)

Ask

⇢

A(x,t)

x < pt, bid levels x > pt, ask levels

Price pt

Latent BidLatent Ask

BidAsk

Some history…❖ ε-intelligence model: first implementation of this idea, realistic market model

yielding concave impact [B Tóth, Y Lempérière, C Deremble, J de Lataillade, J Kockelkoren, J-P Bouchaud, Phys. Rev. X (2011)]

❖ Its generalizations: analysis of generalization of the model: sqrt impact result is robust. Many issues addressed (impact of limit orders, adaptive order flux) [IM, B Tóth, J-P Bouchaud, Phys. Rev. E. 89 (2014)]

❖ Stylized model: extracted a stylized version of the model in which everything can be computed analytical. Impact is exactly found to be sqrt. [IM, B Tóth, J-P Bouchaud , Phys. Rev. Lett. (2014)]

❖ This model: Bridges the gap among the two approaches: still amenable to analytical calculations, but allows realistic effects to be reincorporated without losing analytical control. [J Donier, J Bonart, IM, J-P Bouchaud, arXiv 1412.0141 (2014)]

Setup

x 2 Z

x

price grid

particles

stochastic dyn.@tPt(b(x), a(x)) = F [Pt(b(x), a(x))]

b(x), a(x) 2 N

Let’s define the dynamics of the system with some care…

Ingredients

A. Drift-diffusion: Orders jump left and right at fixed rate. Idiosyncratic component is D and common component is Vt

(stochastic zero-mean rate).

B. Cancellations: Orders on the book might be canceled (rate ν)

C. Depositions: Orders are deposed at rate λ per unit time on both sides of the book (bids left of price and asks to right)

D. Reactions: Two opposite orders sitting at the same location x can annihilate with rate κ. We consider κ→∞.

We introduce a latent order book model with the following dynamics:

Everything is constant in space, due to reason which will be clear later.

The dynamics

2

di↵usion” model, at least in a region close to the current price where this dynamics becomes universal, i.e. independentof the detailed setting of the model – and hence, as emphasized above, of the detailed micro-structure of the marketand of its high-frequency activity. The reaction-di↵usion model in one dimension posits that two types of particles(called B and A), representing in a financial context the intended orders to buy (bids) and to sell (ask) di↵use ona line and disappear whenever they meet A + B ! ; – corresponding to a transaction. The boundary between theB-rich region and the A-rich region therefore corresponds to the price p

t

. This highly stylized order book modelwas proposed in the late 90’s by Bak et al. [13, 14] but never made it to the limelight because the resulting pricedynamics was found to be strongly mean-reverting on all time scales, at odds with market prices which, after a shorttransient, behave very much like random walks. However, some of us (together with B. Toth, [15]) recently realizedthat the analogue of market impact can be defined and computed within this framework, and was found to obey thesquare-root law exactly.

This opens the door to a fully consistent theoretical model of non-linear impact in financial markets, which wepropose in the present paper. We show how all the previously discussed ingredients can be accommodated in aunifying model for the dynamics of the latent order book that is consistent with price di↵usion. When fluctuationsare neglected (in a sense that will be specified below), the impact of a meta-order can be computed exactly, and isfound to exhibit two regimes: when the execution rate is su�ciently slow, the model becomes identical to the linearpropagator framework proposed in [16, 17], with a bare propagator decaying as the inverse square-root of time. Whenexecution is faster, impact becomes fully non-linear and obeys a non-trivial, closed form integral equation. In the tworegimes, the impact of a meta-order grows as the square-root of the volume, but with a pre-factor that depends on theexecution rate in the slow regime, but becomes independent of it in the fast regime – as indeed suggested by empiricaldata. The model predicts interesting price trajectories when trading is interrupted or reversed, leading to e↵ects thatare observed empirically but impossible to account for within a linear propagator model. We demonstrate that pricesin our model cannot be manipulated, in the sense that any sequence of buy and sell orders that starts and ends with azero position on markets leads to a non-positive average profit. This is a non trivial property of our modelling strategy,which makes it eligible for practical applications. Finally, we discuss how our framework suggests a clear separationbetween “mechanical” price moves (i.e. induced by the impact of random trades) and “informational” price moves(i.e. the impact of any public information that changes the latent supply/demand). This is a key point that allowsus to treat consistently, within the same model, di↵usive prices and memory of the order book – which otherwiseleads to strongly mean-reverting prices (see the discussion in [3, 5, 18]). We discuss in the conclusion some of themany interesting problems that our modelling strategy leaves open – perhaps most importantly, how to consistentlyaccount for order book fluctuations that are presumably at the heart of liquidity crises and market crashes.

II. DYNAMICS OF THE LATENT ORDER BOOK

Our starting point is the zero-intelligence model of Smith et al. [19], reformulated in the context of the latent orderbook in [3] and independently in [20]. We assume that each trading (buy/sell) intention of market participant ischaracterized by a reservation price and a volume.1 In the course of time, the dynamics of intentions can be essentiallyof four types: a) reassessment of the reservation price, either up or down; b) partial or complete cancellation of theintention of buying/selling; c) appearance of new intentions, not previously expressed and finally d) matching of anequal volume of buy/sell intentions, resulting in a transaction at a price that delimits the buys/sells regions, andremoval of these intentions from the latent order book. It is clear that provided very weak assumptions are met:i) the changes in reservation prices are well behaved (i.e. have a finite first and second moment) and short-rangedcorrelated in time; ii) the volumes have a finite first moment, one can establish – in the large scale, low frequency,“hydrodynamic” limit – the following set of partial di↵erential equations for the dynamics of the average buy (resp.sell) volume density ⇢

B

(x, t) (resp. ⇢A

(x, t)) at price level x:

@⇢B

(x, t)@t

= �Vt

@⇢B

(x, t)@x

+ D@2⇢

B

(x, t)@x2

� ⌫⇢B

(x, t) + �⇥(pt

� x)� RAB

(x, t);

@⇢A

(x, t)@t

= �Vt

@⇢A

(x, t)@x

+ D@2⇢

A

(x, t)@x2

| {z }a- Drift-Di↵usion

� ⌫⇢A

(x, t)| {z }b- Cancel.

+�⇥(x� pt

)| {z }c- Deposition

�RAB

(x, t)| {z }d- Reaction

; (1)

1 In fact, each participant may have a full time dependent supply/demand curve with di↵erent prices and volumes, with little change inthe e↵ective model derived below.

3

where the di↵erent terms in the right hand sides correspond to the four mechanisms a-d, on which we elaborate below,and p

t

is the coarse-grained position of the price (i.e. averaged over high frequency noise), defined from the condition

⇢A

(pt

, t)� ⇢B

(pt

, t) = 0. (2)

• a- Drift-Di↵usion: The first two terms model the fact that each agent reassess his/her reservation price x due tomany external influences (news, order flow and price changes themselves, other technical signals, etc.). One cantherefore expect that price reassessments contain both a (random) agent specific part that contributes to thedi↵usion coe�cient 2 D and a common component V

t

that shifts the entire latent order book. This shift is dueto a collective price reassessment due for example to some publicly available information (that could well be thepast transactions themselves). The drift component V

t

is at this stage very general; one possibility that we willadopt below is to think of V

t

as a white noise, such that the price pt

is a random walk. Since the derivation ofthese first two terms and the assumptions made are somewhat subtle, we devote Appendix A to a more detaileddiscussion and alternative models; see in particular Eq. (30).

• b- Cancellations: The third term corresponds to partial or complete cancellation of the latent order, with adecay time ⌫�1 independent of the price level x (but see previous footnote2). Consistent with the idea of acommon information, cancellation could be correlated between di↵erent agents. However, this does not a↵ectthe evolution of the average densities ⇢

B,A

(x, t), while it might play a crucial role for the fluctuations of theorder book, in particular to explain liquidity crises.

• c- Deposition: The fourth term corresponds to the appearance of new buy/sell intentions, modelled by a “rainintensity” � modulated by a step-like function ⇥(u), for example the logistic function that we will use below:

⇥(u) =1

1 + e�2µu

. (3)

In other words, we assume that buy orders mostly appear below the current price pt

and sell orders mostlyappear above p

t

. The detailed shape of ⇥(u) actually turns out to be, to a large extent, irrelevant for thepurpose of the present paper; for simplicity we will choose below µ!1, i.e. ⇥(u > 0) = 1 and ⇥(u < 0) = 0.

• d- Reaction: The last term corresponds to transactions when two orders meet with “reaction rate” ; thequantity R

AB

(x, t) is formally the average of the product of the density of A particles and the density of Bparticles, i.e. R

AB

(x, t) ⇡ ⇢A

(x, t)⇢B

(x, t) + fluctuations. However, the detailed knowledge of RAB

(x, t) willnot a↵ect the following discussion. We will consider in the following the limit ! 1, which corresponds tothe case where latent limit orders close to the transaction price all become “visible” limit orders and are dulyexecuted.

Let us insist that Eqs. (1) only describe the average shape of the latent order book, i.e. fluctuations coming fromthe discrete nature of orders are neglected at this stage: see Fig. 1 for an illustration. In particular, the instantaneousposition of the price pinst.

t

– where the density of buy/sell orders vanishes – has an intrinsic non-zero width even inthe limit ! 1 [21], corresponding to the average distance a � b between the highest buy order x = b and thelowest sell order x = a. The instantaneous price can then be conventionally be defined as pinst.

t

= (a + b)/2, but willin general not coincide with the coarse-grained price p

t

defined by the average shape of the latent order book throughEq. (2). Indeed, as shown in [21], the di↵usion width (i.e. the typical distance between pinst.

t

and pt

) is also nonzero and actually larger than the intrinsic width, but only by a logarithmic factor.

In the following, we will neglect both the intrinsic width and the di↵usion width, which is justified if we focuson price changes much larger than these widths. This is the large scale, low frequency regime where our equationsEqs. (1) are warranted. Formally, Eqs. (1) become valid when the market latent liquidity L (defined below) tends toinfinity, since both the intrinsic width and the di↵usion width vanish as L�1/2 [21].

III. STATIONARY SHAPE OF THE LATENT ORDER BOOK

A remarkable feature of Eqs. (1) is that although the dynamics of ⇢A

and ⇢B

is non-trivial because of the reactionterm (that requires a control of fluctuations, see [21]), due to the exact conservation of #A�#B, the combination

2 In full generality, the di↵usion constant D could depend on the distance |x� p

t

| to the transaction price. We neglect this possibility inthe present version of the model, for reasons that will become clear later. A similar remark applies to the cancellation rate ⌫ as well.

The previous ingredients + a continuous space approximation imply:

and

Remark (I): The difficult term to deal with is RAB(x,t) ≈ ρA(x,t) ρB(x,t), which is non-linear

Remark (II): This definition of price is only meaningful for κ→∞

hb(x)it = ⇢B(x, t)

ha(x)it = ⇢A(x, t)

hb(x)a(x)it = RAB(x, t)

where

defines the average price

Market clearing and ϕ

4

Latent

bid

particle

density

⇢ B(x

,t)

Latent

ask

particle

density

⇢ A(x

,t)

x < pt

, bid levels x > pt

, ask levels

Price pt

Instantaneous density

Average density

FIG. 1: Snapshot of a latent order book in the presence of a meta-order, with bid orders (blue boxes) and ask orders (redboxes) sitting on opposite sides of the price line and subject to a stochastic evolution. The dashed lines show the mean valuesthe order densities ⇢

A,B

(x, t), which are controlled by Eq. (1).

'(x, t) := ⇢B

(x, t)� ⇢A

(x, t) evolves according to a linear equation independent of :

@'(x, t)@t

= �Vt

@'(x, t)@x

+ D@2'(x, t)

@x2

� ⌫'(x, t) + � tanh (µ(pt

� x)) , (4)

where pt

is the solution of '(pt

, t) = 0. This solution is expected to be unique for all t > 0 if it is unique at t = 0 (seealso [20]). Note that Eq. (4), without the drift-di↵usion terms, has recently been obtained as the hydrodynamiclimit of a Poisson order book dynamics in [22].

Introducing bpt

=R

t

0

ds Vs

, the above equation can be rewritten in the reference frame of the latent order booky = x� bp

t

as:3

@'(y, t)@t

= D@2'(y, t)

@y2

� ⌫'(y, t) + � tanh (µ(pt

� bpt

� y)) . (5)

Starting from a symmetric initial condition '(y, t = 0) = �'(�y, t = 0) such that pt=0

= 0, it is clear by symmetrythat p

t

= bpt

at all times. Otherwise, pt

converges to bpt

when t!1 and the stationary solution of Eq. (5) reads, inthe limit µ!1:

'st.

(y 0) =�

⌫[1� e�y] ; '

st.

(y � 0) = �'st.

(�y), (6)

with �2 = ⌫/D. This is precisely the solution obtained in [3] which behaves linearly close to the transaction price.But as emphasized in [3], this linear behaviour in fact holds for any value of µ and, more generally, for a very widerange of models – for example if the appearance of new orders only takes place at y = ±L, as in [15], or else if thecoe�cients D, ⌫ are non-trivial (but su�ciently regular) functions of the distance to the price |y|, etc.

3 One should be careful with the “Ito” term when V

s

is a Wiener noise, which adds a contribution to D, see Appendix 1 and Eq. (31)

The reaction term conserves the difference B-A, as a consequence of market clearing.We exploit it by defining:

so that

'(x, t) = ⇢B(x, t)� ⇢A(x, t)

and price is defined by the condition

'(x, t) = 0

3

where the di↵erent terms in the right hand sides correspond to the four mechanisms a-d, on which we elaborate below,and p

t

is the coarse-grained position of the price (i.e. averaged over high frequency noise), defined from the condition

⇢A

(pt

, t)� ⇢B

(pt

, t) = 0. (2)

• a- Drift-Di↵usion: The first two terms model the fact that each agent reassess his/her reservation price x due tomany external influences (news, order flow and price changes themselves, other technical signals, etc.). One cantherefore expect that price reassessments contain both a (random) agent specific part that contributes to thedi↵usion coe�cient 2 D and a common component V

t

that shifts the entire latent order book. This shift is dueto a collective price reassessment due for example to some publicly available information (that could well be thepast transactions themselves). The drift component V

t

is at this stage very general; one possibility that we willadopt below is to think of V

t

as a white noise, such that the price pt

is a random walk. Since the derivation ofthese first two terms and the assumptions made are somewhat subtle, we devote Appendix A to a more detaileddiscussion and alternative models; see in particular Eq. (30).

• b- Cancellations: The third term corresponds to partial or complete cancellation of the latent order, with adecay time ⌫�1 independent of the price level x (but see previous footnote2). Consistent with the idea of acommon information, cancellation could be correlated between di↵erent agents. However, this does not a↵ectthe evolution of the average densities ⇢

B,A

(x, t), while it might play a crucial role for the fluctuations of theorder book, in particular to explain liquidity crises.

• c- Deposition: The fourth term corresponds to the appearance of new buy/sell intentions, modelled by a “rainintensity” � modulated by a step-like function ⇥(u), for example the logistic function that we will use below:

⇥(u) =1

1 + e�2µu

. (3)

In other words, we assume that buy orders mostly appear below the current price pt

and sell orders mostlyappear above p

t

. The detailed shape of ⇥(u) actually turns out to be, to a large extent, irrelevant for thepurpose of the present paper; for simplicity we will choose below µ!1, i.e. ⇥(u > 0) = 1 and ⇥(u < 0) = 0.

• d- Reaction: The last term corresponds to transactions when two orders meet with “reaction rate” ; thequantity R

AB

(x, t) is formally the average of the product of the density of A particles and the density of Bparticles, i.e. R

AB

(x, t) ⇡ ⇢A

(x, t)⇢B

(x, t) + fluctuations. However, the detailed knowledge of RAB

(x, t) willnot a↵ect the following discussion. We will consider in the following the limit ! 1, which corresponds tothe case where latent limit orders close to the transaction price all become “visible” limit orders and are dulyexecuted.

Let us insist that Eqs. (1) only describe the average shape of the latent order book, i.e. fluctuations coming fromthe discrete nature of orders are neglected at this stage: see Fig. 1 for an illustration. In particular, the instantaneousposition of the price pinst.

t

– where the density of buy/sell orders vanishes – has an intrinsic non-zero width even inthe limit ! 1 [21], corresponding to the average distance a � b between the highest buy order x = b and thelowest sell order x = a. The instantaneous price can then be conventionally be defined as pinst.

t

= (a + b)/2, but willin general not coincide with the coarse-grained price p

t

defined by the average shape of the latent order book throughEq. (2). Indeed, as shown in [21], the di↵usion width (i.e. the typical distance between pinst.

t

and pt

) is also nonzero and actually larger than the intrinsic width, but only by a logarithmic factor.

In the following, we will neglect both the intrinsic width and the di↵usion width, which is justified if we focuson price changes much larger than these widths. This is the large scale, low frequency regime where our equationsEqs. (1) are warranted. Formally, Eqs. (1) become valid when the market latent liquidity L (defined below) tends toinfinity, since both the intrinsic width and the di↵usion width vanish as L�1/2 [21].

III. STATIONARY SHAPE OF THE LATENT ORDER BOOK

A remarkable feature of Eqs. (1) is that although the dynamics of ⇢A

and ⇢B

is non-trivial because of the reactionterm (that requires a control of fluctuations, see [21]), due to the exact conservation of #A�#B, the combination

2 In full generality, the di↵usion constant D could depend on the distance |x� p

t

| to the transaction price. We neglect this possibility inthe present version of the model, for reasons that will become clear later. A similar remark applies to the cancellation rate ⌫ as well.

sigmoid deposition

A snapshot of a LOB

Latent

bid

particle

density⇢

B(x,t)

Latent

ask

particle

density⇢

A(x,t)


Price pt


Average density

Instantaneous shape of the book

Average shape of the book

b(x), a(x)

⇢A(x), ⇢B(x)

Average difference bid-ask'(x)

Den

sity

diffe

renc

e'(x,t)


Average '(x, t)

Mechanical reference frameThere is a mean to further simplify the equation. One can reabsorb the drift term with a

suitable choice of coordinates:

pt =

Z t

0ds Vs y = x� pt

so that

4

Latent

bid

particle

density

⇢ B(x

,t)

Latent

ask

particle

density

⇢ A(x

,t)

x < pt

, bid levels x > pt

, ask levels

Price pt


Average density


A,B


'(x, t) := ⇢B

(x, t)� ⇢A


@'(x, t)@t

= �Vt

@'(x, t)@x

+ D@2'(x, t)

@x2

� ⌫'(x, t) + � tanh (µ(pt

� x)) , (4)

where pt



Introducing bpt

=R

t

0

ds Vs


t

as:3

@'(y, t)@t

= D@2'(y, t)

@y2

� ⌫'(y, t) + � tanh (µ(pt

� bpt

� y)) . (5)



t

= bpt


converges to bpt


'st.

(y 0) =�

⌫[1� e�y] ; '

st.

(y � 0) = �'st.

(�y), (6)



s


Remark: Informally, one can interpret this change of coordinates by saying that“information” is factored out…

Stationary behavior

4

Latent

bid

particle

density

⇢ B(x

,t)

Latent

ask

particle

density

⇢ A(x

,t)

x < pt

, bid levels x > pt

, ask levels

Price pt


Average density


A,B


'(x, t) := ⇢B

(x, t)� ⇢A


@'(x, t)@t

= �Vt

@'(x, t)@x

+ D@2'(x, t)

@x2

� ⌫'(x, t) + � tanh (µ(pt

� x)) , (4)

where pt



Introducing bpt

=R

t

0

ds Vs


t

as:3

@'(y, t)@t

= D@2'(y, t)

@y2

� ⌫'(y, t) + � tanh (µ(pt

� bpt

� y)) . (5)



t

= bpt


converges to bpt


'st.

(y 0) =�

⌫[1� e�y] ; '

st.

(y � 0) = �'st.

(�y), (6)



s


The stationary solution is readily found:

for γ2 = ν/D

�/⌫

��/⌫

Stationary

density'

st(x)


Stationary 'st(x)

Locally linear region!

Zooming-in procedure

5

IV. PRICE DYNAMICS WITHIN A LINEAR ORDER BOOK APPROXIMATION (LOBA)

We will therefore, in the following, “zoom” into the universal linear region by taking the formal limit � ! 0 with afixed current

J = D|@y

'st.

|y=0

⌘ �/�. (7)

This current can be interpreted as the volume transacted per unit time in the stationary regime, i.e. the total quantityof buy (or sell) orders that get executed per unit time. As a side remark, it is important to realize that if the driftV

t

contains a Wiener noise component, or jumps, this drift does in fact contribute to J and does not merely shift thelatent order book around without any transactions (see Appendix A).

In the limit ⌫, �! 0 with �/� = J fixed, the stationary solution 'st.

(y) becomes exactly linear:

'st.

(y) = �Jy/D. (8)

This is the regime we will explore in the present paper, although we will comment below on the expected modificationsinduced by non-zero values of ⌫, �. Note that L = J/D ⌘ �

pD/⌫ can be interpreted as the latent liquidity of the

market, which is large when deposition of latent orders is intense (� large) and/or when latent orders have a longlifetime (⌫ small). The quantity L�1 is the analogue, within the LOBA, of “Kyle’s lambda” for a flat order book.

In terms of order of magnitudes, it is reasonable to expect that the latent order book has a memory time ⌫�1

of several hours to several days [3] – remember that we are speaking here of slow actors, not of market makerscontributing to the high-frequency dynamics of the revealed order book. Taking D to be of the order of the pricevolatility, the width of the linear region ��1 is found to be of the order of 1% of the price (see Eq. (6)). Therefore,we expect that our linearized order book approximation (LOBA) will be justified for meta-orders lasting up to severalhours, and impacting the price by less than a fraction of a percent. For larger impacts and/or longer execution times,a more elaborate (and probably less universal) description may be needed.

We now introduce a “meta-order” within our framework and work out in detail its impact on the price. Workingin the reference frame of the unimpacted price bp

t

defined above, we model a meta-order as an extra current of buy (orsell) orders that fall exactly on the transaction price p

t

. Introducing yt

⌘ pt

� bpt

, the corresponding equation for thelatent order book reads, in the LOBA where ⌫, �! 0:

8>><

>>:

@'(y, t)@t

= D@2'(y, t)

@y2

+ mt

�(y � yt

)

@'(y ! ±1, t)@y

= �Ly + o(|y|),(9)

where mt

is the (signed) trading intensity at time t; mt

> 0 corresponding to a buy meta-order. We will assume thatthe meta-order starts at a random time that we choose as t = 0, with no information on the state of the latent orderbook. This means that at t = 0, the order book can be described by its stationary state, '

st.

(y) = �Jy/D. For t > 0,the latent order book is then given by the following exact formula:

'(y, t) = �Ly +Z

t

0

ds msp

4⇡D(t� s)e�

(y�ys)2

4D(t�s) , (10)

where ys

is the transaction price (in the reference frame of the book) at time s, defined as '(ys

, s) ⌘ 0. This leads toa self-consistent integral equation for the price at time t > 0:

yt

=1L

Zt

0

ds msp

4⇡D(t� s)e�

(yt�ys)2

4D(t�s) . (11)

This is the central equation of the present paper, which we investigate in more detail in the next sections.4As a first general remark, let us note that provided impact is small, in the sense that 8t, s, |y

s

� yt

| ⌧ D(t � s),then the above formula exactly boils down to the linear propagator model proposed in [16, 17] (see also [23]), witha square-root decay of impact:

yt

=1L

Zt

0

ds msp

4⇡D(t� s). (12)

4 When m

s

has a non-trivial time dependence, the above equation may not be easy to deal with numerically. It can be more convenientto iterate numerically the Eq. (9) and find the solution of '(y

t

, t) = 0.

5



J = D|@y

'st.

|y=0

⌘ �/�. (7)


t




'st.

(y) = �Jy/D. (8)







t


t

. Introducing yt

⌘ pt

� bpt


8>><

>>:

@'(y, t)@t

= D@2'(y, t)

@y2

+ mt

�(y � yt

)

@'(y ! ±1, t)@y

= �Ly + o(|y|),(9)

where mt



st.


'(y, t) = �Ly +Z

t

0

ds msp

4⇡D(t� s)e�

(y�ys)2

4D(t�s) , (10)

where ys



yt

=1L

Zt

0

ds msp

4⇡D(t� s)e�

(yt�ys)2

4D(t�s) . (11)


s

� yt


yt

=1L

Zt

0

ds msp

4⇡D(t� s). (12)

4 When m

s


t

, t) = 0.

Then you get

�, ⌫ ! 0 L = J/D = �pD/⌫

Remark (I): A precise framework in which the linear approx becomes exact (λ large vol, ν slow dyn)

Let’s consider the regime in which

Stationary

density'

st(x)


Stationary 'st(x)

Locally linear region!

J J

J is the effective current injected at boundaries

Remark (II): J fixes number ofreactions/transactions per unit time!

Results

Meta-ordersWe define a meta-order in this framework by adding a term:

5



J = D|@y

'st.

|y=0

⌘ �/�. (7)


t




'st.

(y) = �Jy/D. (8)







t


t

. Introducing yt

⌘ pt

� bpt


8>><

>>:

@'(y, t)@t

= D@2'(y, t)

@y2

+ mt

�(y � yt

)

@'(y ! ±1, t)@y

= �Ly + o(|y|),(9)

where mt



st.


'(y, t) = �Ly +Z

t

0

ds msp

4⇡D(t� s)e�

(y�ys)2

4D(t�s) , (10)

where ys



yt

=1L

Zt

0

ds msp

4⇡D(t� s)e�

(yt�ys)2

4D(t�s) . (11)


s

� yt


yt

=1L

Zt

0

ds msp

4⇡D(t� s). (12)

4 When m

s


t

, t) = 0.

Representing an extra-flux of orders falling exactly on the mid-price.We also assume:

Q =

Zdsms

mt = 0 for t < 0

total volume executed

stationary book at t=0

Numerical resultsWe find…

x

pt

⇢B ⇢A

Fig. Evolution in presence of a meta-order mt

[fig. by J.Donier]

Numerical resultsWe find…

x

pt

⇢B

Fig. Evolution in presence of a meta-order mt

[fig. by J.Donier]

⇢A

Modified propagator model

5



J = D|@y

'st.

|y=0

⌘ �/�. (7)


t




'st.

(y) = �Jy/D. (8)







t


t

. Introducing yt

⌘ pt

� bpt


8>><

>>:

@'(y, t)@t

= D@2'(y, t)

@y2

+ mt

�(y � yt

)

@'(y ! ±1, t)@y

= �Ly + o(|y|),(9)

where mt



st.


'(y, t) = �Ly +Z

t

0

ds msp

4⇡D(t� s)e�

(y�ys)2

4D(t�s) , (10)

where ys



yt

=1L

Zt

0

ds msp

4⇡D(t� s)e�

(yt�ys)2

4D(t�s) . (11)


s

� yt


yt

=1L

Zt

0

ds msp

4⇡D(t� s). (12)

4 When m

s


t

, t) = 0.

5



J = D|@y

'st.

|y=0

⌘ �/�. (7)


t




'st.

(y) = �Jy/D. (8)







t


t

. Introducing yt

⌘ pt

� bpt


8>><

>>:

@'(y, t)@t

= D@2'(y, t)

@y2

+ mt

�(y � yt

)

@'(y ! ±1, t)@y

= �Ly + o(|y|),(9)

where mt



st.


'(y, t) = �Ly +Z

t

0

ds msp

4⇡D(t� s)e�

(y�ys)2

4D(t�s) , (10)

where ys



yt

=1L

Zt

0

ds msp

4⇡D(t� s)e�

(yt�ys)2

4D(t�s) . (11)


s

� yt


yt

=1L

Zt

0

ds msp

4⇡D(t� s). (12)

4 When m

s


t

, t) = 0.

This behavior can be controlled analytically. The book satisfies the equation:

This yields an exact integral equation for the price trajectory

Remark: This generalizes the propagator model (recovered for small perturbations)!

init. cond perturbation

Absence of price manipulationThis modified propagator satisfies nice properties concerning price manipulation.

Consider a trajectory mt and consider

Cost of a closed loop:

8<

:

C =R T0 dsmsys

0 =R T0 dsms

Then you have

C =1

2

Z T

0

Z T

0dsds0ms M(s, s0)ms0 � 0

because M(s,s’) = K K† is a non-negative operator.

Exact solution for the impactWe have a closed-form solution for constant mt = m0:

yt = ApDt

where the coefficient A satisfies the integral equation

6

This linear approximation is therefore valid for very small trading rates ms

, but breaks down for more aggressiveexecutions, for which a more precise analysis is needed. An ad-hoc non-linear generalisation of the propagator modelwas suggested by Gatheral [23], but is di�cult to justify theoretically. We believe that Eq. (11) above is the correctway to generalize the propagator model, such that all known empirical results can be qualitatively accounted for.

Note that one can in fact define a volume dependent “bid” (or “ask”) price y±t

(q) for a given volume q as thesolution of:

Zyt

y

�t (q)

dy '(y, t) = �Z

y

+t (q)

yt

dy '(y, t) = q. (13)

Clearly, in the equilibrium state, and for q small enough, y±t

(q) = yt

±p

2q/L. After a buy meta-order, however, wewill find that strong asymmetries can appear.

1

0.1 1 10 100 1000

I(Q

)(D

Q/J

)�1/2

m0

/J

0.1

1

10

100

0.1 1 10 100 1000

I(Q

)(D

T)�

1/2

m0

/J

A (m0

/J)�1/2

p2

(m0

/J)1/2

Ap2m

0

/Jm

0

/J

FIG. 2: Left: Dependence of the ratio A/

pm0/J upon the trading rate parameter m0/J . (This ratio coincides with the

empirically used Y ratio if �

2 is identified with D and V with J). The curve interpolates between ap

m0/J dependenceobserved at small trading trading rates and an asymptotically constant regime ⇡

p2 for large m0/J . This is consistent with

the weak dependence of Y upon the trading rate observed in CFM empirical data. Right: Dependence of the impact I(Q) onQ for a fixed execution time T – i.e. a variable m0 = Q/T . Note the crossover between a linear behaviour at small Q and asquare-root behaviour for large Q.

V. THE SQUARE-ROOT IMPACT OF META-ORDERS

The simplest case where a fully non-linear analysis is possible is that of a meta-order of size Q executed at a constantrate m

0

= Q/T for t 2 [0, T ]. In this case, it is straightforward to check that ys

= Ap

Ds is an exact solution ofEq. (11), where the constant A is the solution of the following equation:

A =m

0

J

Z1

0

dup4⇡(1� u)

e�A2(1�

pu)

4(1+p

u) . (14)

- Impact is exactly square root- We have found a non-trivial prefactor

adimensional trading rate

Remarks on the exact solution❖ The dependence in t1/2 is

generic (scale invariance!): [see IM, B Toth, J-P Bouchaud (2014)]

❖ Impact for large executed volumes is independent of the trading rate.

❖ Impact as a function of volume is ∼ Q1/2

❖ You recover the Y ratio (for D ∼ σ2/Tday and J ∼ V/Tday)

6

This linear approximation is therefore valid for very small trading rates ms

, but breaks down for more aggressiveexecutions, for which a more precise analysis is needed. An ad-hoc non-linear generalisation of the propagator modelwas suggested by Gatheral [23], but is di�cult to justify theoretically. We believe that Eq. (11) above is the correctway to generalize the propagator model, such that all known empirical results can be qualitatively accounted for.

Note that one can in fact define a volume dependent “bid” (or “ask”) price y±t

(q) for a given volume q as thesolution of:

Zyt

y

�t (q)

dy '(y, t) = �Z

y

+t (q)

yt

dy '(y, t) = q. (13)

Clearly, in the equilibrium state, and for q small enough, y±t

(q) = yt

±p

2q/L. After a buy meta-order, however, wewill find that strong asymmetries can appear.

1

0.1 1 10 100 1000

I(Q

)(D

Q/J

)�1/2

m0

/J

0.1

1

10

100

0.1 1 10 100 1000

I(Q

)(D

T)�

1/2

m0

/J

A (m0

/J)�1/2

p2

(m0

/J)1/2

Ap2m

0

/Jm

0

/J

FIG. 2: Left: Dependence of the ratio A/

pm0/J upon the trading rate parameter m0/J . (This ratio coincides with the

empirically used Y ratio if �

2 is identified with D and V with J). The curve interpolates between ap

m0/J dependenceobserved at small trading trading rates and an asymptotically constant regime ⇡

p2 for large m0/J . This is consistent with

the weak dependence of Y upon the trading rate observed in CFM empirical data. Right: Dependence of the impact I(Q) onQ for a fixed execution time T – i.e. a variable m0 = Q/T . Note the crossover between a linear behaviour at small Q and asquare-root behaviour for large Q.

V. THE SQUARE-ROOT IMPACT OF META-ORDERS

The simplest case where a fully non-linear analysis is possible is that of a meta-order of size Q executed at a constantrate m

0

= Q/T for t 2 [0, T ]. In this case, it is straightforward to check that ys

= Ap

Ds is an exact solution ofEq. (11), where the constant A is the solution of the following equation:

A =m

0

J

Z1

0

dup4⇡(1� u)

e�A2(1�

pu)

4(1+p

u) . (14)

I(Q) =

A

rJ

m0

!✓Q

L

◆1/2

Generic trading ratesThe property of independence with respect to the trajectory is much more general

than that! Consider mt large. Then

8

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

I(Q

,t)(

DT

)�1/2

A�

1

t/T

0.1

1

1 10 100t/T

m0

/J = 0.1m

0

/J = 1m

0

/J = 10

(t/T )�1/2

FIG. 3: Impact I(Q, t) as a function of rescaled time for various trading rate parameters m0/J . The initial growth of theimpact follows exactly a square-root law, and is followed by a regime shift suddenly after end of the meta-order. While fort = T

+ the slope of the impact function becomes infinite, at large times one observes an inverse square relaxation ⇠p

T/t

with an m0/J dependent pre-factor. Note that the curves for m0/J = 0.1 and m0/J = 1 are nearly indistinguishable.

VII. PRICE TRAJECTORY AT LARGE TRADING INTENSITIES

Our general price equation Eq. (11) is amenable to an exact treatment in the large trading intensity limit mt

� J ,provided m

t

does not change sign and is a su�ciently regular function of time. In such a case, the change of price islarge and therefore justifies a saddle-point estimate of the integral appearing in Eq. (11). This leads to the followingasymptotic equation of motion:

Lyt

|yt

| ⇡ mt

1 + D

✓3

yt

y3

t

� 2m

t

mt

y2

t

◆+ O

✓J2

m2

◆�. (19)

When mt

keeps a constant sign (say positive), the leading term of the above expansion therefore yields the followingaverage impact trajectory:

yt

⇡

s2L

Zt

0

ds ms

, (20)

i.e. a price impact that only depends on the total traded volume, but not on the execution schedule. This is a strongerresult than the one obtained above, where impact was found to be independent of the trading intensity for a uniformexecution scheme. This path independence is in qualitative agreement with empirical results obtained at CFM. UsingEq. (19), systematic corrections to the above trajectory can be computed.

(perturbative expansionin powers of J/m)

which at first order gives (for mt positive)

8

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

I(Q

,t)(

DT

)�1/2

A�

1

t/T

0.1

1

1 10 100t/T

m0

/J = 0.1m

0

/J = 1m

0

/J = 10

(t/T )�1/2



T/t




� J ,provided m

t


Lyt

|yt

| ⇡ mt

1 + D

✓3

yt

y3

t

� 2m

t

mt

y2

t

◆+ O

✓J2

m2

◆�. (19)

When mt


yt

⇡

s2L

Zt

0

ds ms

, (20)


generalizing previous formula for the impact.

Impact decay8

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

I(Q

,t)(

DT

)�1/2

A�

1

t/T

0.1

1

1 10 100t/T

m0

/J = 0.1m

0

/J = 1m

0

/J = 10

(t/T )�1/2



T/t




� J ,provided m

t


Lyt

|yt

| ⇡ mt

1 + D

✓3

yt

y3

t

� 2m

t

mt

y2

t

◆+ O

✓J2

m2

◆�. (19)

When mt


yt

⇡

s2L

Zt

0

ds ms

, (20)


What happens after the trade?

- Right after: Steep decay of impact

I(Q,T + ✏)

I(Q)! 1�

r✏

T

- Long after: Slow relaxation

We control analytically both limits!

I(Q, t)

I(Q)! 1

4

r1

2⇡

m0T

Jt

Permanent impactI didn’t speak about permanent impact yet. How to account for informed trades?

hVsms0i / C(s, s0)

correlation among drift and executed volume

This models trades containing/producing information (e.g., alpha).

Itot

(Q, t > T )

=

I(Q, t > T )� m0�

⇣ (1� e�⇣T )e�⇣(t�T ) + �Q

Example: Let’s assume C(s,s’) = Γ e-ζ (s-s’) . Then we get that

Permanent impact!

Permanent vs transient impact12

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

I(Q

,t)(

DT

)�1/2

A�

1

t/T

0.1

1

1 10 100t/T

FIG. 7: The figure illustrates the relative roles of mechanical and informational impact in determining the price trajectoryduring and after the execution of a meta-order. We have chosen in particular the set of parameters D = J = ⇣ = m = T = 1and � = 0.1. The figure indicates that the mechanical part of the impact is the dominating e↵ect at small times. Thepermanent, informational component of the impact becomes relevant only after the slow decay of the mechanical component,as shown the inset. From a theoretical point of view, the permanent component is important since it counterbalances potentialmarket-making/mean-reversion profits coming from the confining e↵ect of the latent order book on the price.

X. POSSIBLE EXTENSIONS AND OPEN PROBLEMS

The LOBA framework presented above is surprisingly rich and accounts for many empirical observations, but itcan only be a first approximation of a more complex reality. First, we have neglected e↵ects that are in principlecontained in Eqs.(1) but that disappear in the limit of slow latent order books ⌫T ⌧ 1 (such that the memory time1/⌫ is much longer than the meta-order) and large liquidity L, such that the meta-order only probes the linear regionof the book. Re-integrating these e↵ects perturbatively is not di�cult; for example, one finds that the impact I(Q)of a meta-order of size Q, executed at a constant rate, is lowered by a quantity proportional to ⌫T when ⌫T ⌧ 1. Inthe opposite limit ⌫T � 1, one expects that I(Q) becomes linear in Q, since impact must become additive in thatlimit (see [3]). Any other large scale regularisation of the model will lead to the same conclusion. One also expectsthat the deposition, with rate �, of new orders behind the moving price should reduce the asymmetry of impact whenthe trade is reversed. We leave a more detailed calculation of these e↵ects for later investigations.

Another important line of research is to understand the corrections to the LOBA induced by fluctuations, that areof two types: first, as discussed in section II, the theory presented here only deals with the average order book '(x, t),from which the price p

t

is deduced using the definition '(pt

, t) = 0, that allowed us to compute the average impactof a meta-order. However, one should rather compute the impact from the instantaneous definition of the price pinst.

t

(that takes into account the fluctuations of the order book) and then take an average that would lead to I(Q). Thenumerical simulations shown in [15] suggest however that the approximation used here is quite accurate for longmeta-orders, which is indeed expected as the di↵erence |pinst.

t

� pt

| becomes small compared to I(Q).Second, we have assumed that the rest of the market is in its stationary state and does not contribute to the source

term modelling the meta-order. One should rather posit that the flow of meta-order ms

has a random component

Itot

(Q, t)

Iperm(Q, t)I(Q, t)

Relative contributions of permanentand transient impact

Permanent impactdominates at large times!

Slow growth atsmall times!

Discussion

Open problems

❖ The role of fluctuations: the mechanical fluctuations are strongly mean-reverting, and they cross-over at larger times with diffusion due to drift term. Analytically, this is a hard problem.

❖ Fixing the random drift and its correlations with volume: ideally, one would like ms to be a distribution, endogenized together with the one of Vs.

❖ General study of the modified propagator model.

Extensions

❖ Studying the effect of a distribution on ms

❖ Fat-tailed statistics for Vs

❖ Liquidity drought phenomena: what if orders in the latent order book do not materialize instantly in the real order book?

Conclusions❖ Market impact follows a universal square root law which seems to depend

on few, generic ingredients.

❖ We implement a latent order book model incorporating a one-dimensional notion of price and the market clearing condition

❖ In this setting, we are able to decompose price in a mechanical and an informational component.

❖ The mechanical part of the impact dominates at small times. We exactly calculate it through an exact integral equation generalizing the propagator model, obtaining (robustly) square root impact.

❖ We reproduce many empirical features of impact: trajectory independence, absence of price manipulation, slow relaxation.

References

References

❖ B Tóth, Y Lempérière, C Deremble, J de Lataillade, J Kockelkoren, J-P Bouchaud, Anomalous price impact and the critical nature of liquidity in financial markets, Phys. Rev. X (2011)

❖ IM, B Tóth, J-P Bouchaud, Agent-based models for latent liquidity and concave price impact, Phys. Rev. E. 89 (2014)

❖ IM, B Tóth, J-P Bouchaud, Anomalous impact in reaction-diffusion models, Phys. Rev. Lett. (2014)

❖ J Donier, J Bonart, IM, J-P Bouchaud, A fully consistent, minimal model for non-linear market impact,arXiv 1412.0141 (2014)

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