A fully consistent, minimal model for non-linear market impact
Jonathan Donier, Julius Bonart, Iacopo Mastromatteo, Jean-Philippe Bouchaud
AA fully consistent, minimal model for non-linear market impact
J. Donier,
1, 2
J. Bonart,
1, 3
I. Mastromatteo, and J.-P. Bouchaud
1, 3 Capital Fund Management, 23-25 Rue de l’Universit´e, 75007 Paris, France. Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e Pierre et Marie Curie (Paris 6). CFM-Imperial Institute of Quantitative Finance, Department of Mathematics,Imperial College, 180 Queen’s Gate, London SW7 2RH Centre de Math´ematiques Appliqu´ees, CNRS, UMR7641, Ecole Polytechnique, 91128 Palaiseau, France. (Dated: March 3, 2015)We propose a minimal theory of non-linear price impact based on the fact that the (latent)order book is locally linear, as suggested by diffusion-reaction models and general arguments. Ourframework allows one to compute the average price trajectory in the presence of a meta-order, thatconsistently generalizes previously proposed propagator models. We account for the universallyobserved square-root impact law, and predict non-trivial trajectories when trading is interruptedor reversed. We prove that our framework is free of price manipulation, and that prices can bemade diffusive (albeit with a generic short-term mean-reverting contribution). Our model suggeststhat prices can be decomposed into a transient “mechanical” impact component and a permanent“informational” component.
I. INTRODUCTION
The study of market impact (i.e. the way trading influences prices in financial markets) is arguably among the mostexciting current themes in theoretical finance, with many immediate applications ranging from trading cost modellingto important regulatory issues. What is the meaning of the market price if the very fact of buying (or selling) cansubstantially affect that price? The questions above would on their own justify a strong research activity that datesback to the classic Kyle paper in 1985 [1]. But as often in science, it is the empirical discovery of a genuinely surprisingresult that explains the recent spree of activity on the subject (see e.g. [2–6] and refs. therein). In strong contrastwith the predictions of the Kyle model, market impact appears to be neither linear (in the traded quantity Q ) nor permanent , i.e. time independent [7]. As now firmly established by many independent empirical studies, the averageprice change induced by the sequential execution of a total volume Q (which we call meta-order ) appears to followa sub-linear, approximate √ Q law [2, 3, 5, 8–13]. At the end of the meta-order, impact is furthermore observed todecay (partially or completely) towards the unimpacted price [11–14].Quite strikingly, the square-root law appears to be universal , as it is to a large degree independent of details suchas the type of contract traded (futures, stocks, options, Bitcoin [14]...), the geographical position of the market venue(US, Europe, Asia), the time period (1995 → single orders is non universal and highly sensitive to market micro-structure, the impact of meta-orders appears to be extremely robust against micro-structural changes. For example the rise of high-frequency trading(HFT) in the last ten years seems to have had no effect on its validity (compare Refs. [2, 8] that uses pre-2004 datawith [3, 5, 11] that use post-2007 data). This universality strongly suggests that simple, “coarse-grained” modelsshould be able to reproduce the square-root impact law and other slow market phenomena, while abstracting awayfrom many microscopic details that govern order flow and price formation at high frequencies. This line of reasoningis very similar to many situations in physics, where universal large scale/low frequency laws appear for systems withvery different microscopic behaviour. A well known example is the behaviour of weakly interacting molecules which onlarge length scales can be accurately described by the Navier-Stokes equation, with a single “emergent” parameter (theviscosity) that encodes the microscopic specificities of the system. The Navier-Stokes equation can in fact be derivedeither from the statistical description of the dynamics of molecules, through an appropriate coarse-graining procedure,or from general considerations based on symmetries, conservation laws and dimensional arguments. Along this path,two pivotal ideas have recently emerged. One is the concept of a latent order book [3] that contain the intentions oflow-frequency actors at any instant of time, which may or may not materialize in the observable order book. Indeed,since the square-root impact is an aggregate, low-frequency phenomenon, the relevant object to consider cannot bethe “revealed” order book, which chiefly reflects the activity of high frequency market-makers. Simple orders ofmagnitude confirm that the latent liquidity is much higher than the revealed liquidity: whereas the total daily volumeexchange on a typical stock is around 1 / a r X i v : . [ q -f i n . T R ] M a r of 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-diffusion 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 ) diffuse ona line and disappear whenever they meet A + B → ∅ – corresponding to a transaction. The boundary between the B -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. [15, 16] 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. T´oth, [17]) 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 coarse-grained model for the dynamics of the latent order book that is consistent with price diffusion, with asingle emergent parameter – the market liquidity L , defined below. When fluctuations are neglected (in a sense thatwill be specified below), the impact of a meta-order can be computed exactly, and is found to exhibit two regimes:when the execution rate is sufficiently slow, the model becomes identical to the linear propagator framework proposedin [7, 18], with a bare propagator decaying as the inverse square-root of time. When execution is faster, impactbecomes fully non-linear and obeys a non-trivial, closed form integral equation. In the two regimes, the impact of ameta-order grows as the square-root of the volume, but with a pre-factor that depends on the execution rate in theslow regime, but becomes independent of it in the fast regime – as indeed suggested by empirical data. The modelpredicts interesting price trajectories when trading is interrupted or reversed, leading to effects that are observedempirically but impossible to account for within a linear propagator model. We demonstrate that prices in our modelcannot be manipulated, in the sense that any sequence of buy and sell orders that starts and ends with a zero positionon markets leads to a non-positive average profit. This is a non trivial property of our modelling strategy, whichmakes it eligible for practical applications. Finally, we discuss how our framework suggests a clear separation between“mechanical” price moves (i.e. induced by the impact of random trades) and “informational” price moves (i.e. theimpact of any public information that changes the latent supply/demand). This is a key point that allows us to treatconsistently, within the same model, diffusive prices and memory of the order book – which otherwise leads to stronglymean-reverting prices (see the discussion in [3, 5, 19]). We discuss in the conclusion some of the many interestingproblems that our modelling strategy leaves open – perhaps most importantly, how to consistently account for orderbook 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. [20], reformulated in the context of the latentorder book in [3] and independently in [21]. We assume that each trading (buy/sell) intention of market participant ischaracterized by a reservation price and a volume. 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 differential equations for the dynamics of the average buy (resp.sell) volume density ρ B ( x, t ) (resp. ρ A ( x, t )) at price level x : ∂ρ B ( x, t ) ∂t = − V t ∂ρ B ( x, t ) ∂x + D ∂ ρ B ( x, t ) ∂x − νρ B ( x, t ) + λ Θ( p t − x ) − κR AB ( x, t ); (1-a) ∂ρ A ( x, t ) ∂t = − V t ∂ρ A ( x, t ) ∂x + D ∂ ρ A ( x, t ) ∂x (cid:124) (cid:123)(cid:122) (cid:125) a- Drift-Diffusion − νρ A ( x, t ) (cid:124) (cid:123)(cid:122) (cid:125) b- Cancel. + λ Θ( x − p t ) (cid:124) (cid:123)(cid:122) (cid:125) c- Deposition − κR AB ( x, t ) (cid:124) (cid:123)(cid:122) (cid:125) d- Reaction ; (1-b) In fact, each participant may have a full time dependent supply/demand curve with different prices and volumes, with little change inthe effective model derived below. where the different 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 ρ A ( p t , t ) − ρ B ( p t , t ) = 0 . (1) • a- Drift-Diffusion:
The first two terms model the fact that each agent reassess his/her reservation price x dueto many external influences (news, order flow and price changes themselves, other technical signals, etc.). Onecan therefore expect that price reassessments contain both a (random) agent specific part that contributes tothe diffusion coefficient D and a common component V t that shifts the entire latent order book. This shift isdue to a collective price reassessment due for example to some publicly available information (that could wellbe the past transactions themselves). The drift component V t is at this stage very general; one possibility thatwe will adopt below is to think of V t as a white noise, such that the price p t is a diffusive random walk. Sincethe derivation of these first two terms and the assumptions made are somewhat subtle, we devote Appendix Ato a more detailed discussion 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 ν − independent of the price level x (but see previous footnote ). Consistent with the idea of acommon information, cancellation could be correlated between different agents. However, this does not affectthe evolution of the average densities ρ 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” λ modulated by an arbitrary increasing function Θ( u ), expressing that buy orders mostly appearbelow the current price p t and sell orders mostly appear above p t . The detailed shape of Θ( u ) actually turnsout to be, to a large extent, irrelevant for the purpose of the present paper (see Appendix B for details); forsimplicity we will choose below a step function, Θ( u >
0) = 1 and Θ( u <
0) = 0. • d- Reaction:
The last term corresponds to transactions when two orders meet with “reaction rate” κ ; thequantity R AB ( x, t ) is formally the average of the product of the density of A particles and the density of B particles, i.e. R AB ( x, t ) ≈ ρ A ( x, t ) ρ B ( x, t ) + fluctuations. However, the detailed knowledge of R AB ( x, t ) willnot affect the following discussion. We will consider in the following the limit κ → ∞ , which corresponds to thecase where latent limit orders close to the transaction price all become instantaneously visible limit orders thatare duly executed against incoming market orders.Let us insist that Eqs. (1-a,b) 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 p inst. t – where the density of buy/sell orders vanishes – has an intrinsic non-zero width even inthe limit κ → ∞ [22], corresponding to the average distance a − b between the highest buy order x = b and the lowestsell order x = a . The instantaneous price can then be conventionally be defined as p inst. t = ( a + b ) /
2, but will ingeneral not coincide with the coarse-grained price p t defined by the average shape of the latent order book throughEq. (1). Indeed, as shown in [22], the diffusion width (i.e. the typical distance between p inst. t and p t ) is also non zeroand actually larger than the intrinsic width, but only by a logarithmic factor.In the following, we will neglect both the intrinsic width and the diffusion width, which is justified if we focus onprice changes much larger than these widths. This is the large scale, low frequency regime where our coarse-grainedequations Eqs. (1-a,b) are warranted. Formally, Eqs. (1-a,b) become valid when the market latent liquidity L (definedbelow) tends to infinity, since both the intrinsic width and the diffusion width vanish as L − / [22]. In full generality, the diffusion 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: see Appendix B. A similar remark applies to the cancellationrate ν as well. We indeed assume that latent orders become instantaneously visible when close to p inst. t , in such a way that the latent order book andthe observable order book become identical at the best limits. This is of course needed to identify p inst. t with the “real” mid-price. It isvery interesting to ask what happens if the conversion speed between latent orders and real orders is not infinitely fast, or when marketorders become out-sized compared to the prevailing liquidity. As we discuss in the conclusion, this is a potential mechanism for crashes,and the simple coarse-grained framework discussed here has to be adapted to deal with these situations. L a t e n t b i dp a r t i c l e d e n s i t y ρ B ( x , t ) L a t e n t a s k p a r t i c l e d e n s i t y ρ A ( x , t ) x < p t , bid levels x > p t , ask levelsPrice p t Instantaneous densityAverage 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 ρ A,B ( x, t ), which are controlled by Eqs. (1). III. STATIONARY SHAPE OF THE LATENT ORDER BOOK
A remarkable feature of Eqs. (1-a,b) is that although the dynamics of ρ A and ρ B is non-trivial because of thereaction term (that requires a control of fluctuations, see [22]) the combination ϕ ( x, t ) := ρ B ( x, t ) − ρ A ( x, t ) evolvesaccording to a linear equation independent of κ : ∂ϕ ( x, t ) ∂t = − V t ∂ϕ ( x, t ) ∂x + D ∂ ϕ ( x, t ) ∂x − νϕ ( x, t ) + λ sign ( p t − x ) , (2)where p t is the solution of ϕ ( p t , t ) = 0. This solution is expected to be unique for all t > t = 0 (seealso [21]). Note that Eq. (2), without the drift-diffusion terms, has recently been obtained as the hydrodynamic limitof a Poisson order book dynamics in [23].Introducing (cid:98) p t = (cid:82) t d s V s , the above equation can be rewritten in the reference frame of the latent order book y = x − (cid:98) p t as: ∂ϕ ( y, t ) ∂t = D ∂ ϕ ( y, t ) ∂y − νϕ ( y, t ) + λ sign ( p t − (cid:98) p t − y ) . (3)Starting from a symmetric initial condition ϕ ( y, t = 0) = − ϕ ( − y, t = 0) such that p t =0 = (cid:98) p t =0 = 0, it is clear bysymmetry that the equality p t = (cid:98) p t is a solution at all times, since all terms in the above equation are odd when y → − y . For more general initial conditions, p t converges to (cid:98) p t when t → ∞ and the stationary solution of Eq. (3)reads, in the limit µ → ∞ : ϕ st. ( y ≤
0) = λν [1 − e γy ] ; ϕ st. ( y ≥
0) = − ϕ st. ( − y ) , (4) The disappearance of κ can be traced to the conservation of A − B for each reaction A + B → ∅ . One should be careful with the “Ito” term when V s is a Wiener noise, which adds a contribution to D , see Appendix A and Eq. (31) with γ = ν/D . This is precisely the solution obtained in [3] which behaves linearly close to the transaction price.But as emphasized in [3] and in Appendix B, this linear behaviour in fact holds for a very wide range of models – forexample if the appearance of new orders only takes place at some arbitrary boundary y = ± L , as in [17], or else ifthe coefficients D, ν are non-trivial (but sufficiently regular) functions of the distance to the price | y | , etc. IV. PRICE DYNAMICS WITHIN A LOCALLY LINEAR ORDER BOOK (LLOB)
We will therefore, in the following, “zoom” into the universal linear region by taking the formal limit γ → J = D | ∂ y ϕ st. | y =0 ≡ λ/γ. (5)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 drift V 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 ν, λ → λ/γ = J fixed, the stationary solution ϕ st. ( y ) becomes exactly linear: ϕ st. ( y ) = − Jy/D. (6)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 ν, λ . Note that L = J/D ≡ λ (cid:112) D/ν can be interpreted as the latent liquidity of themarket, which is large when deposition of latent orders is intense ( λ large) and/or when latent orders have a longlifetime ( ν small). The quantity L − is the analogue, within a LLOB, 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 ν − 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 γ − is found to be of the order of 1% of the price (see Eq. (4)). Therefore,we expect that restricting the analysis to the linear region of the order book will be justified for meta-orders lastingup to several hours, and impacting the price by less than a fraction of a percent. For larger impacts and/or longerexecution 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 (cid:98) p 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 y t ≡ p t − (cid:98) p t , the corresponding equation for thelatent order book reads, within a LLOB that precisely holds when ν, λ → ∂ϕ ( y, t ) ∂t = D ∂ ϕ ( y, t ) ∂y + m t δ ( y − y t ) ∂ϕ ( y → ±∞ , t ) ∂y = −L , (7)where m t is the (signed) trading intensity at time t ; m t > D , ν and λ ), so that L is a fixed parameter in the above equation. Of course, this assumption might break downwhen the meta-order is out-sized, leading to a sudden increase of the cancellation rate ν and a corresponding drop ofthe liquidity L , which might in turn result in a crash (see the discussion in the conclusion).We will now consider a meta-order that starts at a random time that we choose as t = 0, with no information onthe state of the latent order book. This means that at t = 0, there is no conditioning on the state of the order bookthat can be described by its stationary shape, ϕ st. ( y ) = − Jy/D . For t >
0, the latent order book is then given bythe following exact formula: ϕ ( y, t ) = −L y + (cid:90) t d s m s (cid:112) πD ( t − s ) e − ( y − ys )24 D ( t − s ) , (8)where y s is the transaction price (in the reference frame of the book) at time s , defined as ϕ ( y s , s ) ≡
0. This leads toa self-consistent integral equation for the price at time t > y t = 1 L (cid:90) t d s m s (cid:112) πD ( t − s ) e − ( yt − ys )24 D ( t − s ) . (9)This is the central equation of the present paper, which we investigate in more detail in the next sections. As a first general remark, let us note that provided impact is small, in the sense that ∀ t, s , | y s − y t | (cid:28) D ( t − s ),then the above formula exactly boils down to the linear propagator model proposed in [7, 18] (see also [24]), with asquare-root decay of impact: y t = 1 L (cid:90) t d s m s (cid:112) πD ( t − s ) . (10)This linear approximation is therefore valid for very small trading rates m s , 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 [24], but is difficult to justify theoretically (and leads to highly singular optimal tradingschedules in the continuous time limit [25]). We believe that Eq. (9) above is the correct way to generalize thepropagator 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: (cid:90) y t y − t ( q ) d y ϕ ( y, t ) = − (cid:90) y + t ( q ) y t d y ϕ ( y, t ) = q. (11)Clearly, in the equilibrium state, and for q small enough, y ± t ( q ) = y t ± (cid:112) q/ L . After a buy meta-order, however, wewill find that strong asymmetries can appear.10 . I ( Q )( D Q / J ) − / m /J . . I ( Q )( D T ) − / m /JA ( m /J ) − / √ m /J ) / A p m /Jm /J FIG. 2: Left: Dependence of the ratio A/ (cid:112) m /J upon the trading rate parameter m /J . (This ratio coincides with theempirically used Y ratio if σ is identified with D and V with J ). The curve interpolates between a (cid:112) m /J dependenceobserved at small trading trading rates and an asymptotically constant regime ≈ √ m /J . This is consistent withthe weak dependence of Y upon the trading rate observed in CFM empirical data. Right: Dependence of the impact I ( Q ) on Q for a fixed execution time T – i.e. a variable m = Q/T . Note the crossover between a linear behaviour at small Q and asquare-root behaviour for large Q . 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. (7) and find the solution of ϕ ( y t , t ) = 0. 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 = Q/T for t ∈ [0 , T ]. In this case, it is straightforward to check that y s = A √ Ds is an exact solution ofEq. (9), where the constant A is the solution of the following equation: A = m J (cid:90) d u (cid:112) π (1 − u ) e − A −√ u )4(1+ √ u ) . (12)It is easy to work out the asymptotic behaviour of A in the two limits m (cid:28) J and m (cid:29) J . In the first case, onefinds A ≈ m /J √ π , while in the second case A ≈ (cid:112) m /J . The impact I of a meta-order of size Q , is defined as: I ( Q ) = (cid:104) ε · ( p t + T − p t ) | Q (cid:105) , (13)where (cid:104) . . . | Q (cid:105) denotes an average over all meta-orders of sign ε and volume Q , executed over the time interval [ t, t + T ].We assuming for now that the meta-order is uninformed, in the following sense: (cid:104) ε · (ˆ p t + T − ˆ p t ) | Q (cid:105) = 0 , (14)such that the only contribution is the “mechanical” impact on the dynamics of y t . The case of informed meta-orderswill be treated in Sect. IX. The mechanical impact at the end of the meta-order is then given by y T = A √ DT , i.e.: I ( Q ) = A √ m (cid:112) DQ ≈ (cid:114) m Jπ × (cid:114) Q L ( m (cid:28) J ); I ( Q ) ≈ (cid:114) Q L ( m (cid:29) J ) , (15)i.e. precisely a square-root impact law.In fact, the empirical result is often written as I ( Q ) = Y σ (cid:112)
Q/V where σ is the daily volatility and V ≡ JT d = D L T d the daily traded volume ( T d ≡ Y a constant of order unity. Assuming that σ ∝ DT d (which is thecase if D = 0, see Appendix A), we see that Eq. (15) exactly reproduces the empirical result, with Y proportionalto (cid:112) m /J for small trading intensity m and becoming independent of m for larger trading intensity – see Fig. 2. CFM’s empirical data indeed suggests that Y only very weakly depends on the trading intensity, which is nicelyexplained by the present framework. VI. IMPACT DECAY: BEYOND THE PROPAGATOR MODEL
The next interesting question is impact relaxation: how does the price behave after the meta-order has beenexecuted, i.e. when t > T . Mathematically, the impact decay is given by the solution of: y t = Dm J (cid:90) T d s (cid:112) πD ( t − s ) e − ( yt − A √ Ds )24 D ( t − s ) , ( t > T ) (16)In the small m /J limit, the linear propagation model is appropriate and predicts the following impact relaxation: I ( Q, t > T ) I ( Q ) = √ t − √ t − T √ T , (17)that behaves as 1 − (cid:112) ( t − T ) /T very shortly after the end of the meta-order and as (cid:112) T /t/ m /J is more subtle, in particular at short times. The full analysis is givenin Appendix C and reveals that the rescaled initial decay of impact is, quite unexpectedly, still exactly given byEq. (17), independently of m /J . For large times, y t →
0, which implies that asymptotically | y t − A √ Ds | (cid:28) √ Dt , Note that I ( Q ) is a slight abuse of notations since the impact in fact depends in general in the whole trajectory m s . The results in the two limits are (up to prefactors) those obtained in [17] within an explicit reaction-diffusion setting. Note that in agreement with our interpretation of the latent order book, the quantity JT must be interpreted as the volume of“slow” orders executed in a time T , removing all fast intra-day activity that averages out and therefore cannot withstand (other thantemporarily) the incoming meta-order. . . . .
81 0 1 2 3 4 5 I ( Q , t )( D T ) − / A − t/T .
111 10 100 t/Tm /J = 0 . m /J = 1 m /J = 10( t/T ) − / FIG. 3: Impact I ( Q, t ) as a function of rescaled time for various trading rate parameters m /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 for t = T + the slope of the impact function becomes infinite, at large times one observes an inverse square relaxation ∼ (cid:112) T /t with an m /J dependent pre-factor. Note that the curves for m /J = 0 . m /J = 1 are nearly indistinguishable. i.e. the exponential term in Eq. (16) is approximately equal to one, leading to an asymptotic rescaled impact decay as (cid:112) m T / πJt/
4. We plot in Fig. 3 the normalized free decay of impact for different values of m /J for the “mid-price” p t , and in Fig. 4 the corresponding evolution of the effective “bid-ask” p ± t ( q ) for a given volume q , illustrating howthe latent order book becomes more and more asymmetric as m /J increases.The above analysis can be extended to the case where trading is reverted after time T , i.e. m t = m for t ∈ [0 , T ]and m t = − m for t ∈ [ T, T ]. This case is particularly interesting since it puts the emphasis on the lack of liquiditybehind the price for large execution rates. Within the linear propagator approximation, it is easy to show that thetime needed for the price to come back to its initial value (before continuing to be pushed down by the sell meta-order)is given by T /
4. In the non linear regime m (cid:29) J , the price goes down much faster, and reaches its initial valueafter a time given by JT / m (cid:28) T /
VII. PRICE TRAJECTORY AT LARGE TRADING INTENSITIES
Our general price equation Eq. (9) is amenable to an exact treatment in the large trading intensity limit m t (cid:29) J ,provided m t does not change sign and is a sufficiently 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. (9). This leads to the following00 .
51 0 1 2 3 4 m / J = t/T . m / J = . m / J = Price change I ( Q, t )( DT ) − / A − BidAskPriceBidAskPriceBidAskPrice
FIG. 4: Evolution in time of the bid p − t ( q ) (blue line) and the ask p + t ( q ) (red line) while executing a meta-order at a rate m /J ∈ { , , } . The price p t (green line) is also shown for comparison. The three curves correspond to the execution of aconstant volume Q = m T , while the threshold q has been set by q = 10 − Q . The plot illustrates how a large execution rate m /J induces a locally asymmetric liquidity profile around the price, see also Fig. 5. asymptotic equation of motion: L y t | ˙ y t | ≈ m t (cid:20) D (cid:18) y t ˙ y t − m t m t ˙ y t (cid:19) + O (cid:18) J m (cid:19)(cid:21) ; (18)see Appendix D for details of the derivation and for the next order term, of order J /m .When m t keeps a constant sign (say positive), the leading term of the above expansion therefore yields the followingaverage impact trajectory: y t ≈ (cid:115) L (cid:90) t d s m s , (19)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. (18), systematic corrections to the above trajectory can be computed (see Appendix D). Perhaps surprisingly,the execution cost of a given quantity Q is found to be independent of the trading schedule even to first-order in J/m – see Appendix D for a proof. Exploring the optimal execution schedule within the full non-linear price equationEq. (9), and comparing the results with those obtained in Ref. [25], is left for a future study.0 − . − . . . . . − − . . D ϕ / J xm /J = 1 − − − − xm /J = 10 t = 0 . t = 0 . t = 1 t = 0 . t = 0 . t = 1 FIG. 5: Evolution of the order book shape ϕ ( x, t ) during the execution of a meta-order at small trading rate m /J = 1 (leftplot) and large trading rate m /J = 10 (right plot). The solid lines indicate the profile of the book at t = 0 .
01 (green line), t = 0 . t = 1 (blue line). While the displacement of the mid-price follows a square root law, the function Dϕ ( x, t ) /J + x satisfies a scaling relation determined by the parameter m /J – see also Appendix C and Fig. 8. VIII. ABSENCE OF PRICE MANIPULATION
We now turn to a very important issue, that of price manipulation. Although not proven to be impossible in reality,it looks highly implausible that one will ever be able to build a money machine that “mechanically” pumps money outof markets. Any viable model of price impact should therefore be such that mechanical price manipulation, leadingto a positive profit after a closed trading loop, is impossible in the absence of information about future prices [27]. Here, we show that the non-linear price impact model defined by Eq. (9) is free of price manipulation, generalizingthe result of [28] for the linear propagator model, see also [29]. We start by noticing that the average cost of a closedtrajectory is given by: C = (cid:90) T d s m s y s , with (cid:90) T d s m s = 0 (20)and y s given by Eq. (9). The above formula simply means that the executed quantity m s d s between time s and s + d s is at price y s . Because the initial and final positions are assumed to be zero, there is no additional marked-to-marketboundary term. Using Eq. (9), it is not difficult to show that C can be identically rewritten as a quadratic form: C = 12 (cid:90) T (cid:90) T d s d s (cid:48) m s M ( s, s (cid:48) ) m s (cid:48) , (21) Note that property is highly important for practical purposes as well, since using an impact model with profitable closed tradingtrajectories in – say – dynamical portfolio algorithms would lead to instabilities. The alert reader might wonder whether m s is really the executed quantity, rather than the submitted quantity, as the definition of m s as a flux of buy/sell orders suggest. However, within the present framework where m s is deposited precisely at the mid-price p t , onecan check that in the limit κ → ∞ , and provided latent and real liquidity are the same close to p t , the opposite flow of limit ordersimmediately adapts to absorb exactly the incoming meta-order. − − . .
51 0 0 . . I ( Q , T )( D T ) − / A − t/Tm /J = 0 m /J = 1 m /J = 10 m /J = 1000 m /J = ∞ FIG. 6: Trajectory of the average price before and after a sudden switch of the sign of a meta-order. We have considered m t = m for t < T and m t = − m for t > T , and plotted the expected price change as a function of time for different valuesof m . The curves for finite m (solid lines) are also compared with the theoretical benchmark m = 0, corresponding to thepropagator model (dotted line), and to the m = ∞ limit (dot-dashed line). We find that non-linear effects in the large m regime makes to propagator approximation invalid, and increase considerably the impact of the reversal trade. where M ( s, s (cid:48) ) is a non-negative operator, since it can be written as a sum of “squares” KK † , or more precisely: M ( s, s (cid:48) ) = D L (cid:90) ∞−∞ d z z (cid:90) + ∞−∞ d u K z ( s, u ) K ∗ z ( s (cid:48) , u ) , K z ( s, u ) ≡ Θ( s − u ) e − Dz ( s − u )+ izy s . (22)This therefore proves that C ≥ any execution schedule, i.e. price manipulation is impossible within a LLOB (see[6] for loosely related ideas). We note, en passant , that this proof extends to a much larger class of Markovian orderbook dynamics, where the reservation price of latent orders evolves, for example, according to a L´evy process (andnot necessarily a diffusion, as assumed heretofore – see Appendix A).
IX. MECHANICAL VS. INFORMATIONAL IMPACT
We now imagine that the agent executing his/her meta-order has some information about the future price, i.e. thatthe execution flow m t is correlated with the future motion of the latent order book V t (cid:48) for t (cid:48) > t . The apparent impactof the meta-order will now contain two contributions that are, within our framework, additive . Assuming again, forsimplicity, that m t = m , one finds that the average price difference can be written as: (cid:104) ε · ( p t − p ) | Q (cid:105) = (cid:104) ε · ( (cid:98) p t − (cid:98) p ) | Q (cid:105) + (cid:104) ε · ( y t − y ) | Q (cid:105) , (23)where now the first term is non-zero. More explicitly, this leads to: (cid:104) p t − p (cid:105) = m (cid:90) t d s (cid:90) s d s (cid:48) C ( s − s (cid:48) ) + A ( m ) √ Dt, ( t ≤ T ) (24)where C ( s − s (cid:48) ) ∝ (cid:104) V s m s (cid:48) (cid:105) is a measure of the temporal correlation between meta-orders and future collective latentorder moves. Let us insist that we do not assume any causality here: C ( s − s (cid:48) ) can be interpreted either as theinformation content of the order that predicts future price moves (i.e. the so-called “alpha”), or as the collectivereaction of the market to the order flow, i.e. the fact that agents may change their valuation as a result of the tradingitself (see [7, 30] for a discussion of this duality).2The second term in the right hand side of Eq. (24) corresponds to the “mechanical” component of the impactdiscussed above, corresponding to the square-root impact. The first term, on the other hand, may behave verydifferently as a function of T . For example, if C ( s − s (cid:48) ) has a range much smaller than T , the first term is expectedto grow like Q and not √ Q .When t > T , i.e. after the end of the meta-order, the informational contribution adds to the impact decay computedabove and can substantially change the apparent evolution of (cid:104) p t − p (cid:105) . In order to fix ideas, let us assume that C ( s − s (cid:48) ) = Γ ζe − ζ ( s − s (cid:48) ) (other functional forms would not change the qualitative conclusions below). The behaviourof the “total” impact for t > T is then given by: I tot. ( Q, t > T ) = I ( Q, t > T ) + Γ Q − m Γ ζ (1 − e − ζT ) e − ζ ( t − T ) −→ t →∞ Γ Q, (25)which shows that on top of the relaxing mechanical impact (the first term), there is a growing contribution comingfrom the informational content of the trade (or alternatively from the collective reaction of the market to that trade)that saturates at large time to a finite value proportional to Q – see Fig. 7. This corresponds to a “permanent”component of impact. That the permanent component of impact should be linear in Q conforms well with theassumptions of [1, 2]. However, our calculation shows that the empirical determination of the mechanical componentof impact should carefully take into account any possible information content of the analyzed trades, as well as thepossible auto-correlation of the trades. This parallels the discussion offered in [11, 13], where attempts are madeto measure the decay of mechanical impact I ( Q, t > T ) in equity markets, with the conclusion that the mechanicalcomponent of impact seems indeed to relax to zero at large times.The possibility of generating a permanent impact by correlating the collective drift V t with the flow of meta-orders m s is in fact important to make our model internally consistent. Absent the permanent impact component, a randomflow of meta-orders would give rise to a strongly mean-reverting contribution to the price (on top of the random walkcontribution (cid:98) p t = (cid:82) t d s V s ), and therefore potentially profitable mean-reversion/market making strategies. This profitcan however be reduced to zero by increasing the permanent impact component (i.e. the Γ factor above), that actsas an adverse selection bias for market makers. On this point, see the discussion in [31]. X. POSSIBLE EXTENSIONS AND OPEN PROBLEMS
The LLOB framework presented above is surprisingly rich and accounts for many empirical observations, but it canonly be a first approximation of a more complex reality. First, we have neglected effects that are in principle containedin Eqs. (1-a,b) but that disappear in the limit of slow latent order books νT (cid:28) /ν is much longer than the meta-order) and large liquidity L , such that the meta-order only probes the linear region ofthe book. Re-integrating these effects perturbatively is not difficult; for example, one finds that the impact I ( Q ) of ameta-order of size Q , executed at a constant rate, is lowered by a quantity proportional to νT when νT (cid:28)
1. In theopposite limit νT (cid:29)
1, one expects that I ( Q ) becomes linear in Q , since impact must become additive in that limit(see [3]). Any other large scale regularisation of the model will lead to the same conclusion. One also expects thatthe deposition, with rate λ , of new orders behind the moving price should reduce the asymmetry of impact when thetrade is reversed. We leave a more detailed calculation of these effects for later investigations.Another important line of research is to understand the corrections to the LLOB induced by fluctuations, thatare of two types: first, as discussed in section II, the theory presented here only deals with the average order book ϕ ( x, t ), from which the price p t is deduced using the definition ϕ ( p t , t ) = 0, that allowed us to compute the averageimpact of a meta-order. However, one should rather compute the impact from the instantaneous definition of theprice p inst. t (that takes into account the fluctuations of the order book) and then take an average that would lead to I ( Q ). The numerical simulations shown in [17] suggest however that the approximation used here is quite accuratefor long meta-orders, which is indeed expected as the difference | p inst. t − p t | 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 sourceterm modelling the meta-order. One should rather posit that the flow of meta-order m s has a random componentthat adds to the particular meta-order that one is particularly interested in. There again, a calculation based onthe average order book is not sufficient, since the interaction with other uncorrelated meta-orders then triviallydisappears. Following [22], one finds that random fluctuations in m s do contribute to a strongly mean-reverting termin the variogram of the price trajectory, that should be taken into account in a consistent way. Interestingly, thisgeneric mean-reverting component leads to an excess short-term volatility that is commonly observed in financialmarkets; more quantitative work on that front would therefore be worthwhile.Finally, other extensions/modifications may be important in practice: as noted above, the cancellation rate ν isexpected to increase with the intensity of meta-orders. Furthermore, the incoming flow of latent orders λ and/or the300 . . . .
81 0 1 2 3 4 5 I ( Q , t )( D T ) − / A − t/T .
111 10 100 t/T
FIG. 7: The figure illustrates the relative rˆoles 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 = ζ = m = T = 1and Γ = 0 .
1. The figure indicates that the mechanical part of the impact is the dominating effect 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 effect of the latent order book on the price. lifetime of orders 1 /ν can be expected to be increasing functions of the distance to the price | x − p t | , i.e. better pricesshould attract more, and more patient buyers (or sellers), in such a way that the latent order book becomes convex at large distances. This would naturally explain why all impact data known to us appear to grow even slower than √ Q at large Q ([32, 33], and CFM, unpublished data). Another interesting path would be to allow the “drift” term V t in Eq. (1-a,b) to become non-Gaussian and thereby study a cumulant expansion of the square-root law. XI. CONCLUSION
In this paper, we have proposed a minimal theory of non-linear price impact based on a linear (latent) orderbook approximation, inspired by diffusion-reaction models and general arguments. As emphasized in [3, 5, 17], ourmodelling strategy does not rely on any equilibrium or fair-pricing conditions, but rather relies on purely statisticalconsiderations. Our approach is strongly bolstered by the universality of the square-root impact law, in particular onthe Bitcoin market – as recently documented in [14] – where fair-pricing arguments are clearly unwarranted becauseimpact is much smaller than trading fees.Our framework allows us to compute the average price trajectory in the presence of a meta-order, that consistentlygeneralizes previously proposed propagator models. Our central result is the dynamical Eq. (9), which not onlyreproduces the universally observed square-root impact law, but also predicts non-trivial trajectories when tradingis interrupted or reversed. Quite surprisingly, we find that the short time behaviour of the free decay of impactis identical to that predicted by a propagator model, whereas the impact of a reversed trade is found to be muchstronger. The latter result is in qualitative agreement with empirical observations [26]. We have shown that our model4is free of price manipulation, which makes it the first consistent, non-linear and time dependent theory of impact. Oursetting also suggests how prices can be naturally decomposed into a transient “mechanical impact” component and apermanent “informational” component, as initially proposed by Almgren et al. [2], and recently exploited in [11, 13]– see Section IX. Let us insist once again that this decomposition allowed us to construct diffusive prices (albeit witha generic short-term mean-reverting contribution). Although our calculations are based on several approximations (restricting to a locally linear order book andneglecting fluctuations), we believe that it provides a sound starting point for further extensions where the neglectedeffects can be progressively reinstalled. Of particular importance is the potential feedback loop between price moves,order flow and the shape of the latent and of the revealed order books. In particular, we have assumed that latentorders instantaneously materialize in the real order book as the distance to the price gets small: any finite conversiontime might however contribute to liquidity droughts, in particular when prices accelerate, leading to an unstablefeedback loop. As emphasized in [3, 35, 36] this might be triggered by the anomalous liquidity fluctuations inducedby the vanishingly small liquidity in the vicinity of the price. This mechanism could explain the universal power-lawdistribution of returns that appear to be unrelated to exogenous news but rather to unavoidable, self-induced liquiditycrises.
Acknowledgments
We warmly thank M. Abeille, R. Benichou, X. Brokmann, J. de Lataillade, C. Deremble, J. D. Farmer, J. Gatheral,J. Kockelkoren, C. A. Lehalle, Y. Lemp´eri`ere, F. Lillo, E. S´eri´e and in particular M. Potters and B. T´oth for manydiscussions and collaborations on these issues. We also thank P. Blanc, N. Kornman, T. Jaisson, M. Rosenbaum andA. Tilloy for useful remarks on the manuscript. One of us (IM) benefited from the support of the “Chair Marketsin Transition”, under the aegis of “Louis Bachelier Finance and Sustainable Growth” laboratory, a joint initiative of´Ecole Polytechnique, Universit´e d’´Evry Val d’Essonne and F´ed´eration Bancaire Fran¸caise.
Appendix A: Derivation of the drift/diffusion term
In order to give more flesh to the microscopic assumptions underlying the drift/diffusion equation written in Eqs. (1-a,b), let us assume first that each agent contributes to a negligible fraction of the latent order book, which is probablya good approximation for deep liquid markets. A model for thin markets, where some participants contribute to asubstantial fraction of the liquidity, is discussed below, but leads to a very similar final result.Between t and t + δt , each agent i revises its reservation price p i to p i + β i ξ t + η i,t , where ξ t is common to all i representing some public information (news, but also the price change itself or the order flow, etc.) and β i > i to the news, which we imagine to be a random variable from agent to agent, with a pdf Π( β )mean normalized to [ β i ] i = 1. [[ ... ] i represents a cross-sectional average over agents.] Some agents may over-react,others under-react; β i might in fact be itself time dependent, but we assume that the distribution of β ’s is stationary.The completely idiosyncratic contribution η i,t is an independent random variable both across different agents and intime, with distribution R ( η ) of mean zero and rms Σ. We assume that within each price interval x, x + d x lie latentorders from a large number of agents. The density of latent orders ρ ( x, t ) therefore evolves according to the followingMaster equation: ρ ( x, t + δt ) = (cid:90) ∞ d β Π( β ) (cid:90) ∞−∞ d ηR ( η ) (cid:90) d yρ ( y, t ) δ ( x − y − βξ t − η ) , (26)or: ρ ( x, t + δt ) = (cid:90) ∞ d β Π( β ) (cid:90) ∞−∞ d ηR ( η ) ρ ( x − βξ t − η, t ) . (27)Assuming that the price revisions βξ t + η over a small time interval δt are small enough, a second-order expansionKramers-Moyal of the above equation leads to (see [37] for an in-depth discussion of this procedure): ρ ( x, t + δt ) − ρ ( x, t ) = − ξ t ρ (cid:48) ( x, t ) + 12 (cid:0) [ β ] ξ t + Σ (cid:1) ρ (cid:48)(cid:48) ( x, t ) + . . . (28) A clear theoretical justification of the square-root impact law is also important if one wants to promote the idea of impact discountedmark-to-market accounting rules, as advocated in [34]. ξ t = V t δt and Σ = 2 D δt in which case the continuous time limitreads: ∂ρ ( x, t ) ∂t = − V t ∂ρ ( x, t ) ∂x + D ∂ ρ ( x, t ) ∂x (29)or that ξ t = V t √ δt , where V t is now a Gaussian white noise of variance σ , and again Σ = 2 Dδt , in which case thecontinuous time limit should be written as:d ρ ( x, t ) = − d W t ∂ρ ( x, t ) ∂x + D d t ∂ ρ ( x, t ) ∂x (30)with d W t a Wiener noise and D ≡ D + [ β ] σ /
2, to wit, the diffusion constant involves both the idiosyncraticcomponent and the dispersion of reaction to random information. This is the interpretation we will mostly follow inthe present paper. A careful derivation of the corresponding equation in the reference frame of the price (cid:98) p t = (cid:82) t d W s finally gives the diffusion part of Eq. (3) in the main text: ∂ρ ( y, t ) ∂t = D ∂ ρ ( y, t ) ∂y ; D ≡ D + σ (cid:90) d β Π( β )( β − ; (31)i.e. only the dispersion of reaction β − t and t + d t , a fraction φ ∈ [0 ,
1] (possiblytime dependent) of agents collectively change their price estimate by an amount d W t , with no other idiosyncraticcomponent. This leads to:d ρ ( x, t ) = φ [ ρ ( x − d W t , t ) − ρ ( x, t )] = − φ d W t ρ (cid:48) ( x, t ) + 12 φσ d tρ (cid:48)(cid:48) ( x, t ) , (32)that essentially corresponds to the case above with Π( β ) = (1 − φ ) δ ( β ) + φδ ( β − (cid:98) p t = (cid:82) t φ d W s , one finds Eq. (31) with D ≡ φ (1 − φ ) σ /
2. Note that, clearly, these collective price revisions must bythemselves induce transactions whenever 0 < φ < ρ ( x, t ) should include jumps in thecontinuous time limit, i.e. one would find an integro-differential equation rather than a partial differential equationfor ρ ( x, t ). However, if the jump process is homogeneous in space, one can diagonalize the evolution operator inFourier space. This allows one to show that price manipulation is impossible in that case as well. Appendix B: A generically linear latent order book
Let us consider the case where the deposition flow is not constant. This leads to the following equation for thestationary state of the latent order book:
D ∂ ϕ st. ( y ) ∂y − νϕ st. ( y ) + λ (Θ( y ) − Θ( − y )) = 0 , (33)with ϕ st. ( y ) = − ϕ st. ( − y ) (and in particular the market clearing condition ϕ st. ( y = 0) = 0).Let us assume that Θ( y ) − Θ( − y ) behaves, for y → ∞ , as a constant that we can set to unity. The solution ϕ st. ( y )then converges to λ/ν for large y , so we set: ϕ st. ( y ) = λν + Ψ( y ) , (34)where Ψ( y ) = 1 − Θ( y ) + Θ( − y ) with Ψ( y → ∞ ) →
0, and:
D ∂ Ψ( y ) ∂y − ν Ψ( y ) = λ Ξ( y ) , (35)where Ξ( y → ∞ ) →
0. The boundary condition on Ψ( y ) at large y means that we can look at a solution of the form:Ψ( y ) = ψ ( y ) e − √ ν/D y , (36)6so that: Dψ (cid:48)(cid:48) ( y ) − √ νDψ (cid:48) ( y ) = λ Ξ( y ) e √ ν/D y . (37)Finally, one finds: ϕ st. ( y ) = λν (cid:104) − e − √ ν/D y (cid:105) + λD e − √ ν/D y (cid:90) y d y (cid:48) e √ ν/D y (cid:48) (cid:90) ∞ y (cid:48) d y (cid:48)(cid:48) e − √ ν/D y (cid:48)(cid:48) Ξ( y (cid:48)(cid:48) ) . (38)The solution given in the main text corresponds to Ξ ≡
0, so that only the first term survives. From the above explicitform, one sees that provided the integral (cid:82) ∞ d y (cid:48)(cid:48) e − √ ν/Dy (cid:48)(cid:48) Ξ( y (cid:48)(cid:48) ) is finite, the behaviour of ϕ st. ( y ) is always linearin the vicinity of y = 0. Only a highly singular the deposition rate, diverging faster than 1 /y when y →
0, wouldjeopardize the local linearity of the latent order book (see also [38], where this property was first discussed).
Appendix C: Shape of the order book during constant rate execution and initial relaxation of impact
When the trading rate is a constant (equal to m ), one can exhibit an exact scaling solution of the time dependentorder book of the form ϕ ( x, t ) = m (cid:112) t/D F ( x √ Dt ), where F is the solution of:2 F (cid:48)(cid:48) ( u ) + uF (cid:48) ( u ) − F ( u ) = − δ ( u − A ) . (39)As we show below, this equation can be solved and gives the exact shape of the book at t = T , from which the initialrelaxation (after trading has stopped) can be deduced.Writing F = uG in the above equation, one finds a first order linear equation for H = G (cid:48) : H (cid:48) ( u ) + (cid:18) u u (cid:19) H ( u ) = 1 u δ ( u − A ) (40)which is easily solved as: H ( u ) = H u e − u / , ( u < A ); H ( u ) = H − Ae A / u e − u / , ( u > A ) . (41)There are two boundary conditions that are useful. One is the very definition of the price position, x = A √ Dt or A = u , for which φ ( x, t ) = x or F ( A ) = AJ/m . The second remark is that when u = 0, the integral defining F canbe computed, leading to F (0) = A √ π e A / (cid:90) ∞ A / d vv / e − v . (42)This allows one to fix H since G (cid:48) ( u ) = F (cid:48) (0) /u − F (0) /u ≈ − F (0) /u when u →
0, to be compared with H ( u ) ≈ H /u in the same limit. Hence H = − F (0).The final solution for F ( u ) is easily obtained from integrating H and multiplying by u (see Fig. 8). Using G ( A ) = J/m , one finds: G ( u ) = F (0) (cid:90) Au d vv e − v / + J/m , ( u ≤ A ) (43)and G ( u ) = − ( F (0) + Ae A / ) (cid:90) uA d vv e − v / + J/m , ( u ≥ A ) (44)Of special interest is the slope of F for u = A ± ; with F (cid:48) = uH ( u ) + G ( u ) one finds: F (cid:48) ( A − ) = J/m − F (0) A e − A / ; F (cid:48) ( A + ) = J/m − F (0) A e − A / − . (45)700 . . . . . . − − − − D ϕ / J + x xm /J = 1 00 . . . . − − − xm /J = 10 t = 0 . t = 0 . t = 1 t = 0 . t = 0 . t = 1 FIG. 8: Evolution of the average order book ϕ ( x, t ), represented for two different values of the perturbation parameter m /J .The curves in different colors are snapshots taken at different times of the difference between the perturbed and the unperturbedaverage of the book. The scaling form of the book ϕ ( x, t ) = m (cid:112) t/DF ( x √ Dt ) is clear from the plots, which illustrate how aflat region of the book is formed at large values of m /J . Now, shortly after the meta-order has stopped, one can look at the solution of the diffusion equation in the vicinityof the final price p T = A √ DT , using a piece-wise linear function for the initial condition, with slopes given by F (cid:48) ( A ± ).The solution then reads, with t − T = ∆ small: ϕ ( x, t ) = m √ T ∆ (cid:20) F (cid:48) ( A − ) z + ( F (cid:48) ( A + ) − F (cid:48) ( A − )) (cid:90) ∞ z d u ( u − z ) √ π e − u / (cid:21) , (46)with z = ( p T − x ) / √ D ∆. The position of the price is given by p t = p T − √ D ∆ z ∗ , with z ∗ such that: F (cid:48) ( A − ) z ∗ + ( F (cid:48) ( A + ) − F (cid:48) ( A − )) (cid:90) ∞ z ∗ d u ( u − z ∗ ) √ π e − u / = 0 . (47)Using the expression for F (0) and the result above for F (cid:48) ( A ± ), this equation simplifies to: z ∗ (cid:90) ∞ A / d vv / e − v = 2 (cid:90) ∞ z ∗ d u ( u − z ∗ ) e − u / . (48)Changing variables in the RHS from u to v = u /
4, and integrating by parts, one finds: z ∗ (cid:90) ∞ A / d vv / e − v = z ∗ (cid:90) ∞ z ∗ / d vv / e − v , (49)which leads to z ∗ = A for all m /J . [The other solution, z ∗ = 0, is spurious].Hence, the initial stage of the impact relaxation can be written in a super-universal way: p t ≈ t → T + p T (cid:34) − (cid:114) t − TT (cid:35) , (50)i.e. exactly the result from the propagator model, even in the non-linear regime!8 Appendix D: A saddle point approximation for large trading rates
In this Appendix we develop a systematic procedure in order to solve perturbatively Eq. (9). The first step in orderto find a solution is to introduce an expansion parameter (cid:15) (cid:28)
1, which we use in order to control the amplitude ofthe trading rate through m t → m t (cid:15) − . Such substitution implies a scaling of the solution of the form y t → y t (cid:15) − / ,leading to an equation for the price of the form: L y t = (cid:90) t d s m s (cid:112) πD(cid:15) ( t − s ) e − ( yt − ys )24 D(cid:15) ( t − s ) . (51)which is equivalent to the one which one would have by leaving invariant m t and by performing the substitutions D → D(cid:15) and J → J(cid:15) . Hence, the large trading regime is equivalent to the one of slow diffusion.In this case, it is evident that the integral is dominated by times s close to t , which suggests to Taylor expand boththe trading rate m s and the price y s around s = t , so to insert the resulting series in the integral appearing in Eq. (9).The dominating term results m t (cid:90) ∞ d u √ πDu(cid:15) e − ˙ y t u D(cid:15) = m t | ˙ y t | − . (52)The successive corrections to above result can be computed systematically, as they involve developing the square andthe exponential function in the Gaussian term in Eq. (9). In particular, by exploiting the identity (cid:90) ∞ d u e − z u u α = Γ(1 + α ) | z | − α ) (53)it is possible to derive the expansion L y t | ˙ y t | = m t (cid:20) D(cid:15) ) (cid:18) y t ˙ y t − m t m t ˙ y t (cid:19) (54)+ ( D(cid:15) ) (cid:18) m t ˙ y t −
30 ˙ m t ¨ y t ˙ y t − m t ... y t ˙ y t + 45 m t ¨ y t m t ˙ y t (cid:19) + 5( D(cid:15) ) (cid:18) − m t ˙ y t + 42 ¨ m t ¨ y t ˙ y t + 28 ˙ m t ... y t ˙ y t + 7 m t .... y t ˙ y t m t ˙ y t (cid:19) + 5( D(cid:15) ) (cid:18) −
168 ˙ m t ¨ y t ˙ y t − m t ¨ y t ... y t ˙ y t + 252 m t ¨ y t m t ˙ y t (cid:19) + O ( (cid:15) ) (cid:21) , whose first-order terms match the form reported in Eq. (18). Each of the contributions of order (cid:15) n can be seenequivalently either as suppressed by the small value of the diffusion constant diffusion (through a D n factor) or bythe large value trading rate (through a factor of the order of m − nt ).Finally, note that the implicit equation above needs to be inverted in order to obtain a relation yielding y t as afunction of m t . This is possible by using Eq. (54) as an iterative scheme for y t , which allows to calculate L y t | ˙ y t | = m t (55)+ ( J(cid:15) ) (cid:18) − Q t ˙ m t m t (cid:19) + ( J(cid:15) ) (cid:18) − Q t ˙ m t m t − s ˙ m t m t + 4 ˙ m t m t (cid:90) d s Q s ˙ m s m s + 16 Q t ˙ m t m t − Q t ¨ m t m t (cid:19) + O ( (cid:15) ) , where Q t = (cid:82) t d s m s .As a simple application of the above formula, consider the case where m t ≥ , ∀ t ∈ [0 , T ]. In this case, ˙ y t is also nonnegative and we can remove the absolute value in the above equation. To order (cid:15) , the solution of the above equationis (assuming y = 0): 12 L y t = Q t + J(cid:15) ( − t − Q t m t ) , (56)9where we have used integration by parts. Now, the cost C ( Q ) associated to buying a total quantity Q in time T isgiven by: C ( Q ) = (cid:90) T d s m s y s , Q = (cid:90) T d s m s . (57)Therefore, to order (cid:15) : C ( Q ) = (cid:114) L (cid:90) T d s m s (cid:112) Q s − J(cid:15) √ L (cid:90) T d s (cid:20) (cid:112) Q s + sm s √ Q s (cid:21) . (58)After integrating by parts the last term, one finally finds that the impact cost is, to order (cid:15) , independent of the tradingschedule m s , and given by: C ( Q ) = 23 (cid:114) L Q / (cid:20) − J(cid:15)T Q (cid:21) . (59)The correction term is negative, as expected since a slower trading speed leaves time for the opposing liquidity todiffuse towards the traded price. [1] A. Kyle, Continuous auctions and insider trading , Econometrica: Journal of the Econometric Society, 1315-1335 (1985).[2] R. Almgren, C. Thum, E. Hauptmann, H. Li,
Direct estimation of equity market impact , Risk, 18(7), 5762 (2005).[3] B. T´oth, Y. Lemp´eri`ere, C. Deremble, J. De Lataillade, J. Kockelkoren, J.-P. Bouchaud,
Anomalous price impact and thecritical nature of liquidity in financial markets , Physical Review X, 1(2), 021006 (2011).[4] J. D. Farmer, A. Gerig, F. Lillo, and H. Waelbroeck,
How efficiency shapes market impact , Quantitative Finance, 13, 1743(2013).[5] I. Mastromatteo, B. T´oth, J.-P. Bouchaud,
Agent-based models for latent liquidity and concave price impact
Phys. Rev. E89, 042805 (2014).[6] I. Skachkov,
Market Impact Paradoxes , Wilmott, 2014, 7178 (2014).[7] J.-P. Bouchaud, J.D. Farmer, F. Lillo,
How markets slowly digest changes in supply and demand , in Handbook of FinancialMarkets: Dynamics and Evolution, Elsevier: Academic Press, 57-156 (2008).[8] N. Torre, M. Ferrari,
Market impact model handbook , BARRA Inc., Berkeley (1997), available at , but see also [9].[9] R. C. Grinold, R. N. Kahn,
Active Portfolio Management (New York: The McGraw-Hill Companies, Inc., 1999).[10] E. Moro, J. Vicente, L. G. Moyano, A. Gerig, J. D. Farmer, G. Vaglica, F. Lillo, R. N. Mantegna,
Market impact andtrading profile of hidden orders in stock markets.
Physical Review E, 80(6), 066102 (2009).[11] C. Gomes, H. Waelbroeck,
Is Market Impact a Measure of the Information Value of Trades? Market Response to Liquidityvs Informed Trades , Quantitative Finance (2014), DOI: 10.1080/14697688.2014.963140.[12] N. Bershova, D. Rakhlin,
The Non-Linear Market Impact of Large Trades: Evidence from Buy-Side Order Flow , Quanti-tative Finance 13, 1759-1778 (2013).[13] X. Brokmann, E. S´eri´e, J. Kockelkoren, J.-P. Bouchaud,
Slow decay of impact in Equity markets , arXiv:1407.3390, sub-mitted to Market Microstructure and Liquidity.[14] J. Donier, J. Bonart,
A million metaorder analysis of market impact on the bitcoin , arXiv:1412.4503, submitted to MarketMicrostructure and Liquidity.[15] P. Bak, M. Paczuski, M. Shubik,
Price variations in a stock market with many agents , Physica A: Statistical Mechanicsand its Applications, 246(3), 430-453 (1997).[16] L. H. Tang, G. S. Tian,
Reaction-diffusion-branching models of stock price fluctuations.
Physica A: Statistical Mechanicsand its Applications, 264(3), 543-550 (1999).[17] I. Mastromatteo, B. T´oth, J.-P. Bouchaud,
Anomalous impact in reaction-diffusion models , Phys. Rev. Lett. 113, 268701(2014).[18] J.-P. Bouchaud, Y. Gefen, M. Potters, M. Wyart,
Fluctuations and response in financial markets: the subtle nature of“random” price changes.
Quantitative Finance, 4(2), 176-190 (2004).[19] D. E. Taranto, G. Bormetti, F. Lillo,
The adaptive nature of liquidity taking in limit order books , J. Stat. Mech. (2014)P06002.[20] E. Smith, J. D. Farmer, L. Gillemot, and S. Krishnamurthy,
Statistical theory of the continuous double auction , Quantit.Financ. 3, 481 (2003).[21] C.-A. Lehalle, O. Gu´eant, and J. Razafinimanana,
High frequency simulations of an order book: a Two-Scales approach ,in F. Abergel, B.K. Chakrabarti, A. Chakraborti, and M. Mitra (Eds.), Econophysics of Order-Driven Markets, NewEconomic Windows. Springer (2010). [22] G.T. Barkema, M.J. Howard, J.L. Cardy, Reaction-diffusion front for A + B → in one dimension , Physical Review E,53(3), R2017 (1996).[23] X. Gao, J. G. Dai, A. B. Dieker, and S. J. Deng, Hydrodynamic limit of order book dynamics , arXiv:1411.7502.[24] J. Gatheral,
No-dynamic-arbitrage and market impact.
Quantitative Finance, 10, 749-759 (2010).[25] G. Curato, J. Gatheral, F. Lillo,
Optimal execution with nonlinear transient market impact , arXiv:1412.4839.[26] R. Benichou, CFM internal report (2013), and R. Benichou et al., in preparation.[27] G. Huberman, W. Stanzl,
Price Manipulation and Quasi-Arbitrage , Econometrica, , 12471275 (2004).[28] A. Alfonsi, A. Schied, Optimal trade execution and absence of price manipulations in limit order book models
SIAM Journalon Financial Mathematics 1 (1), 490-522 (2010).[29] A. Alfonsi, P. Blanc,
Dynamic optimal execution in a mixed-market-impact Hawkes price model , arXiv:1404.0648.[30] J.-P. Bouchaud,
Price Impact , in Encyclopedia of Quantitative Finance, John Wiley & Sons Ltd. (2010).[31] M. Wyart, J.-P. Bouchaud, J. Kockelkoren, M. Potters, M. Vettorazzo,
Relation between Bid-Ask Spread, Impact andVolatility in Double Auction Markets , Quantitative Finance, 8, 41-57 (2008).[32] E. Zarinelli, M. Treccani, J. D. Farmer, F. Lillo,
Beyond the square root: Evidence for logarithmic dependence of marketimpact on size and participation rate , arXiv:1412.2152.[33] C.-A. Lehalle, private communication, and E. Bacry, A. Iugay, M. Lasnierz, C.-A. Lehalle,
Market impacts and the lifecycle of investors orders , arXiv:1412.0217.[34] F. Caccioli, J.P. Bouchaud, J. D. Farmer,
Impact-adjusted mark-to- market valuation and the criticality of leverage , arXivpreprint 1204.0922, published in Risk Magazine, May 2012.[35] F. Lillo, J.D. Farmer,
The key role of liquidity fluctuations in determining large price fluctuations , Fluctuations and NoiseLetters, 5, L209 (2005).[36] J.-P. Bouchaud,
The Endogenous Dynamics of Markets: Price Impact, Feedback Loops and Instabilities , in Lessons fromthe 2008 Crisis, A. Berd Edt., Risk Publications (2011).[37] C. W. Gardiner,
Stochastic Methods , 4th Edition, Springer, Berlin (2009).[38] J.-P. Bouchaud, M. M´ezard, M. Potters,