Apparent impact: the hidden cost of one-shot trades
AApparent impact: the hidden cost of one-shot trades
Iacopo Mastromatteo
Centre de Math´ematiques Appliqu´ees, CNRS, ´Ecole Polytechnique,UMR 7641, 91128 Palaiseau, France
Abstract
We study the problem of the execution of a moderate size order in an illiquidmarket within the framework of a solvable Markovian model. We suppose that in orderto avoid impact costs, a trader decides to execute her order through a unique trade,waiting for enough liquidity to accumulate at the best quote. We find that despitethe absence of a proper price impact, such trader faces an execution cost arising froma non-vanishing correlation among volume at the best quotes and price changes. Wecharacterize analytically the statistics of the execution time and its cost by mappingthe problem to the simpler one of calculating a set of first-passage probabilities on asemi-infinite strip. We finally argue that price impact cannot be completely avoidedby conditioning the execution of an order to a more favorable liquidity scenario.
Market impact refers to the expected price change after the sequential execution of a givenvolume of contracts in a financial market [1]. It refers to one of the most fundamentalaspects of market microstructure, as it encompasses the information about how a financialmarket reacts to an incoming fluxes of orders, and ultimately allows prices to reflect funda-mental information [2]. Market impact is a central concept also for practitioners, who needto split their large orders (also called meta-orders) in sequences of smaller size child-ordersin order to minimize transaction costs [3].The existence of such order-splitting procedures raises a practical question: Starting fromwhat volume is it appropriate to split a meta-order, and when is it wiser to execute byusing a single trade? Even though the instantly available liquidity is often insufficientin order to instantly execute a large meta-order at the best available price, for moderatesize meta-orders it could be convenient to wait opportunistically until a sufficiently largefluctuation of the outstanding liquidity is realized, so to clear the meta-order at the bestquote with a single trade. While intuitively one would place the threshold of volume a few1 a r X i v : . [ q -f i n . T R ] J un rror bars away from the average value of the outstanding volume, it would be desirableto have a model precisely quantifying how to optimally perform this choice.In this work we analyze such a one-shot execution strategy within a specific solvable frame-work, identifying its average cost and characterizing its statistical fluctuations. Interest-ingly, we find that despite the absence of a proper market impact such a one-shot executionstrategy has a positive average cost. In particular, we find that whenever the outstandingliquidity is correlated with the direction of past price moves, waiting for sufficient volumeto accumulate at the best quote can be expensive.More importantly, we find that such a cost would be detected in data as an extra componentof the slippage (i.e., the expected price change between the decision time and the executiontime of a meta-order) previously neglected in the literature, which is unrelated to thetraditional notion of market impact and that is linked to the microstructural liquidityprofile of the traded contracts. Yet, we dub it apparent impact , as in empirical data itwould appear as a component of the curve describing the transient impact of a meta-orderemerging in absence of a price trend or a short-term alpha, and even in the case in whichno trade at all is performed.The reason why such a component of the impact should be properly taken into accountin the estimation of the impact function is the extremely small entity of the price signal:while the fluctuations of daily returns are typically of the order of a few %, the impactgenerally consists of some basis points, implying a signal to noise ratio around 10 − [2]. Thisforces one to carefully remove even very small sources of bias – such as the aforementionedapparent component of impact – from the raw impact curves, in order not to compromisetheir statistical analysis. In fact, while the most prominent features of market impact (e.g.,its concave dependence upon the executed volume) seem to be roughly universal [4, 5, 6, 7,8, 9, 10, 11, 12, 13], there is no widely-shared consensus about its detailed structure (e.g.,the precise value of the exponent, the dependence of the impact function upon the tradingstrategy and the participation rate).We will focus our attention on a specific scenario in which these ideas can be addressedanalytically. In particular we will consider a stylized version of the framework presentedin [14, 15], addressing the problem of a trader who needs to submit a buy meta-order ofsize Q to the market (in an arbitrary amount of time), and who decides to wait until asufficient amount of liquidity is available at the best quote before executing it by using asingle trade, in order not to incur impact costs. We will be able to argue that: • Between the decision time and the execution time the price will drift on average,even in absence of trend, by an amount I ( Q ), where I ( · ) is a function of the executedvolume, whose precise shape depends upon the details of market microstructure; • Such a price change is not linked to genuine impact (neither mechanical or behav-ioral), as we take as the execution time the instant right before the submission of the2rder, in which the trader has not yet executed her order.Our main conclusion is that past price changes can induce future trading activity simplyfor liquidity reasons: If less liquidity is available at the ask whenever prices have a bullishtrend, then buyers need to be more patient while price moves up. Conversely, they canexecute more quickly when prices decline. We believe that, irrespective of the particularmodel that we have chosen in order to illustrate this behavior, such phenomenon ariseswith greater generality, and that it generates additional execution costs with respect to thetraditional market impact. Our framework should then be regarded as a minimal modelcapturing the emergence of the apparent component of the impact function, allowing toexplore analytically the relation among liquidity at the best quote, local price trend andspeed of execution.The plan of the paper is the following: In Sec. 2 we introduce our market model, reviewingits properties and introducing the notation adopted in the following sections. The mainresults are given in Sec. 3, where we present the order execution procedure for the extratrader and calculate the expected properties of the model at the moment of the execution.Sec. 4 presents a critical assessment of the modeling assumptions, and discusses severalgeneralizations. Finally, we draw our conclusions in Sec. 5. For the sake of clarity, themore technical parts are relegated to the appendices.
We adopt a framework inspired by the one considered in [14, 15] as a background modelof the market. In particular, we consider a Markovian model in which all the informationrelative to the current state of the market is encoded in these variables: • The mid-price x t ; • The volume available at the bid queue V bt and the volume available at the ask queue V at .This choice is motivated by the strong concentration of market activity at the best quotesfor large tick stocks [16], and by the observation that a large component of price variationscan be accounted for by the dynamics at the best quotes [17, 18]. In order to modelthe mechanism of quote revision, we suppose that anytime either queue is exhausted,the volume of both queues is reset and prices are updated accordingly. In particular ifthe ask queue is emptied, we assume that the volumes at the best queues V = ( V b , V a )instantaneously revert to ( V sml , V lrg ), while in the opposite case we assume the new volumesto be ( V lrg , V sml ), which allows to model microscopic mean-reversion of price (see e.g. [18]).3e suppose in the former case the price to jump up by one half of a tick (which we assumewithout loss of generality to be of size w = 2), while in the latter case we assume price todecrease by w/
2. This should be appropriate in the case of large tick contracts, for whichthe spread is closed almost immediately after the exhaustion of the volume at either of thebest queues [19].For the dynamics of the queues we choose a stylized version of the one first consideredin [14], in which we disregard the granularity in the dynamics (the Poissonian nature ofthe volume jumps) and focus on the diffusive limit derived in [15] (see Sec. 4 for a criticalassessment of our assumptions, and a list of possible generalizations of our approach).Within our simplified description of the queues dynamics, the volumes at the best quotesevolve according to the equation ∂P ( t, V b , V a ) ∂t = D ∂ P ( t, V b , V a ) ∂ ( V b ) + D ∂ P ( t, V b , V a ) ∂ ( V a ) + µ ∂P ( t, V b , V a ) ∂V b + µ ∂P ( t, V b , V a ) ∂V b , (1)where we take D = 1 without loss of generality and where we assume µ ≥
0, so to modelqueues whose volume drifts towards zero.Under this dynamics no detailed description of thedistinct type of orders hitting the best queues is provided . Rather, this scenario builds ona coarse-grained description of the volume available at the best quotes which is appropriatewhen volumes sitting at the best queues are sufficiently large.Finally, the procedure of one-shot execution of a buy order illustrated above implies in thiscontext in the presence of a random variable T representing the time elapsed between the decision time t = 0 and the execution time t = T , the first time in which the ask volume V a equals Q . Our goal is to provide a statistical characterization of T , together with theone of the price x t (the number of positive jumps minus the negative ones) and the one forhitting number n t (the number of jumps of either type) at the execution time T . At first we will illustrate the properties of this model in the regime in which no traderis present, and the two queues evolve freely under the dynamics (1) and the boundaryconditions specified above. We use the word freely in order to indicate the unconditionalevolution of the queues (i.e., the absence of the extra-trader), as opposed the one in whichvolumes at the ask cannot do not exceed the value of Q (i.e., when the extra-trader ispresent). In the free case, the coordinates of the system ( V b , V a ) can then diffuse in thewhole positive orthant, reverting to the appropriate state (either ( V lrg , V sml ) or ( V sml , V lrg ))as soon as they touch either of the boundaries. In either case, the variables x t and n t are conveniently updated. We are interested in particular in showing the results for the Notice that in this framework market orders and cancellations are not distinguishable. x t and the hitting number n t . The object that weare required to compute in order to obtain a solution for x t and n t is the probability thata system starting from coordinates ( V b , V a ) hits one of the the two boundaries (either V a = 0 or V b = 0) at a given time t . These quantities will be denoted respectively with p ↑ ( t, V b , V a ) and p ↓ ( t, V b , V a ). It is easy to show (see for instance Ref. [20]) that theirLaplace transforms ˆ p ↑ ( ω, V b , V a ) and ˆ p ↓ ( ω, V b , V a ) satisfy the equations ω ˆ p α ( ω, V b , V a ) = 12 ∂ ˆ p α ( ω, V b , V a ) ∂ ( V b ) + 12 ∂ ˆ p α ( ω, V b , V a ) ∂ ( V a ) − µ ∂ ˆ p α ( ω, V b , V a ) ∂V b − µ ∂ ˆ p α ( ω, V b , V a ) ∂V a , (2)where α ∈ {↑ , ↓} . The difference among p ↑ and p ↓ is encoded in the different boundaryconditions: for V b (cid:54) = 0 one has p ↑ ( t, V b ,
0) = δ t , p ↓ ( t, V b ,
0) = 0, while for V a (cid:54) = 0one has p ↓ ( t, , V a ) = δ t , p ↑ ( t, , V a ) = 0. Equivalently ˆ p ↑ ( ω, V b ,
0) = 1 − ˆ p ↓ ( ω, V b ,
0) = 1,ˆ p ↓ ( ω, , V a ) = 1 − ˆ p ↑ ( ω, , V a ) = 1. The values ¯ p ↑ ( V b , V a ) = ˆ p ↑ (0 , V b , V a ) and ¯ p ↓ ( V b , V a ) =ˆ p ↓ (0 , V b , V a ) obtained for ω = 0 represent the probability of hitting the boundaries atany time between t = 0 and t = ∞ starting from the configuration ( V b , V a ), and verify¯ p ↑ ( V b , V a ) + ¯ p ↓ ( V b , V a ) = 1, indicating that the boundaries are hit almost surely. Oncethat these functions have been calculated (the details about the derivation can be foundin App. A), it is possible to obtain the generating function for the price x t , denoted withΦ x ( ω, s ), as well as the one for the hitting number n t , denoted with Φ n ( ω, s ). They aredefined as Φ x ( ω, s ) = ∞ (cid:88) x = −∞ e − xs (cid:90) ∞ dt e − ωt P x ( t, x ) (3)Φ n ( ω, s ) = ∞ (cid:88) n = −∞ e − ns (cid:90) ∞ dt e − ωt P n ( t, n ) , (4)where P x ( t, x ) and P n ( t, x ) are respectively the probability for the price and for the hittingnumber of taking value x (respectively n ) at time t . By exploiting the Markov property ofthe model, one can prove (App. B) thatΦ x ( ω, s ) = 1 ω (cid:34)(cid:18) − ˆ p ↑↑ − ˆ p ↓↑ − ˆ p ↑↓ − ˆ p ↓↓ (cid:19) T ∞ (cid:88) n =0 (cid:18) ˆ p ↑↑ e − s ˆ p ↑↓ e − s ˆ p ↓↑ e s ˆ p ↓↓ e s (cid:19) n (cid:18) ˆ p ↑ e − s ˆ p ↓ e s (cid:19)(cid:35) (5)+ 1 ω (1 − ˆ p ↑ − ˆ p ↓ ) , Φ n ( ω, s ) = 1 ω (cid:34)(cid:18) − ˆ p ↑↑ − ˆ p ↓↑ − ˆ p ↑↓ − ˆ p ↓↓ (cid:19) T ∞ (cid:88) n =0 (cid:18) ˆ p ↑↑ ˆ p ↑↓ ˆ p ↓↑ ˆ p ↓↓ (cid:19) n (cid:18) ˆ p ↑ ˆ p ↓ (cid:19) e − ( n +1) s (cid:35) (6)+ 1 ω (1 − ˆ p ↑ − ˆ p ↓ ) , p ↑ = ˆ p ↑ ( ω, V b , V a ) ˆ p ↓ = ˆ p ↓ ( ω, V b , V a ) (7)ˆ p ↑↑ = ˆ p ↑ ( ω, V sml , V lrg ) ˆ p ↓↑ = ˆ p ↓ ( ω, V sml , V lrg ) (8)ˆ p ↑↓ = ˆ p ↑ ( ω, V lrg , V sml ) ˆ p ↓↓ = ˆ p ↓ ( ω, V lrg , V sml ) , (9)and ( V b , V a ) are the coordinates from which the system starts at t = 0. Equations (5)and (6) are readily solved by diagonalization of their respective transition matrices, asshown in App. B. In the free case the bid-ask symmetry holds (i.e, p ↑ = p ↓ , p ↑↑ = p ↓↓ and p ↑↓ = p ↓↑ ). We will show in Sec. 3 that the presence of a extra agent passively waiting for avolume Q breaks this symmetry, leading to the emergence of the aforementioned apparentcomponent of the impact.As the the behavior of the model changes qualitatively in the cases µ = 0 and µ >
0, wewill discuss the phenomenology of the model in the two cases separately.
In the driftless case the volumes at the best almost surely drop to zero for any startingcondition (one can in fact verify that the sum ¯ p ↑ + ¯ p ↓ is equal to one). The exhaustion of thevolume can indeed take a very large time, as the volume fluctuations are anomalously large.This leads to a logarithmic sub-diffusion of the price in the long time limit. The assump-tion V sml < V lrg induces short-time mean reversion – in accordance with well-establishedempirical evidence concerning the short-time behavior of the volatility function – partiallydamping the asymptotic value of the price fluctuations. We will illustrate this behavior inthe following section.The absorption probabilities in the driftless scenario can be calculated explicitly (App. A),and their values may be expressed as the integralsˆ p ↑ ( ω, V b , V a ) = 2 π (cid:90) ∞ dk sin( kV a ) k
11 + 2 ω/k (1 − e −√ k +2 ω V b ) (10)ˆ p ↓ ( ω, V b , V a ) = 2 π (cid:90) ∞ dk sin( kV a ) k e −√ k +2 ω V b . (11)The values for ω = 0, expressing the probability of the sign of the next price change, aremore conveniently written in terms of elementary functions, and result¯ p ↑ ( V b , V a ) = 2 π arctan (cid:18) V b V a (cid:19) (12)¯ p ↓ ( V b , V a ) = 2 π arctan (cid:18) V a V b (cid:19) . (13)6he mean absorption time by either of the boundary is infinite (equivalently, the derivativesof Eqs. (10) and (11) with respect to ω are divergent at ω = 0). The explicit expressionof the absorption probability then needs to be inserted into the generating functions (82)and (83) in order to estimate numerically the momenta of the price change and the hittingtimes. A small ω expansion (performed in App. A) allows to evaluate the large time be-havior of these quantities, which results (cid:104) x t (cid:105) = 0 (14) (cid:104) x t (cid:105) − (cid:104) x t (cid:105) = χ t log t (cid:18) π V sml V lrg (cid:19) + o ( t/ log t ) (15) (cid:104) n t (cid:105) = t log t (cid:18) π V sml V lrg (cid:19) + o ( t/ log t ) (16) (cid:104) n t (cid:105) − (cid:104) n t (cid:105) = t log ( t ) (cid:18) π V sml V lrg (cid:19) + o ( t / log t ) , (17)where we have introduced the asymmetry parameter χ = ˆ p ↑↑ / ˆ p ↓↑ = ˆ p ↓↓ / ˆ p ↑↓ accounting forthe short-time mean-reversion of price. An inspection Eq. (15) reveals that the behaviorof price is sub-diffusive, as the variance of price increases less then linearly. This behaviorderives from the broad tails of the distribution for the absorption time, whose anomalouslylarge fluctuations cause the mean absorption time to diverge. Fig. 1 summarizes theseresults by showing the evolution in time of the price change x t and the hitting number n t ,and by comparing it with the results of numerical simulations. µ > The phenomenology of a system in which the drift µ is non-zero is to some extent similarto the one discussed above, as the volumes at the best fall to zero almost surely. The mostnotable difference is that the average absorption time by either of the boundaries is finite.This leads to a diffusive behavior of the price at large times, while at short times the pricehas a mean-reverting behavior for V sml < V lrg similar to the one observed in the driftlesscase.The Laplace transforms of the absorption probabilities for µ > p ↑ ( ω, V b , V a ) = 2 π (cid:90) ∞ dk sin( kV a ) k e µV a ω + µ ) /k (cid:20) − e − (cid:16) √ k +2 ω +2 µ − µ (cid:17) V b (cid:21) (18)ˆ p ↓ ( ω, V b , V a ) = 2 π (cid:90) ∞ dk sin( kV a ) k e µV a µ /k e − (cid:16) √ k +2 ω +2 µ − µ (cid:17) V b . (19)7 h x t i /t ( h x t i − h x t i ) /t t h n t i /tµ = 1 µ = 1 / µ = 0 µ = 1 µ = 1 / µ = 0 µ = 1 µ = 1 / µ = 0Figure 1: Evolution in time of the average price change (cid:104) x t (cid:105) , hitting number (cid:104) n t (cid:105) andvariogram ( (cid:104) x t (cid:105) − (cid:104) x t (cid:105) ) /t for the free diffusion problem. The solid lines indicate theresult of simulations for which we used the set of parameters V = 2 , V sml = 1 , V lrg = 3,while the shaded regions account for two-sigma statistical fluctuations around the meanvalues obtained by averaging over 1000 realizations. The different curves describe boththe driftless case ( µ = 0, corresponding to the red line) and the drifted case ( µ = 1 / µ = 1 are associated respectively with the green and the blue line). The dashed linesindicate the asymptotic predictions displayed in Sec. 2.8hile the probabilities for the sign of the next price change are recovered by setting ω = 0in above formula.The average absorption times can be obtained by differentiation of Eqs. (18) and (19),which lead to convergent integrals, as opposed to the driftless scenario discussed above inwhich the resulting expressions were divergent.These results have been used in Fig. 1 in order to show the large time asymptotics for themean values of x t and n t and their fluctuations in the case µ (cid:54) = 0. In fact the large timebehavior can be written explicitly by expanding the generating functions close to the point ω = 0. We find at leading order in time: (cid:104) x t (cid:105) = 0 (20) (cid:104) x t (cid:105) − (cid:104) x t (cid:105) = t (cid:104) t hit (cid:105) χ + O (1) (21) (cid:104) n t (cid:105) = t (cid:104) t hit (cid:105) + O (1) (22) (cid:104) n t (cid:105) − (cid:104) n t (cid:105) = (cid:18) (cid:104) t hit (cid:105) − (cid:104) t hit (cid:105) (cid:104) t hit (cid:105) (cid:19) t + O (1) , (23)where for q ∈ N we have defined the momenta (cid:104) t qhit (cid:105) = (cid:104) t qhit, ↑ (cid:105) = (cid:104) t qhit, ↓ (cid:105) relative to theaverage time required to hit any of the boundaries starting from either initial condition. Theabove expression allows to relate the (asymptotic) volatility σ = lim t →∞ (cid:104) x t (cid:105) /t = χ/ (cid:104) t hit (cid:105) to the microstructural parameters governing the model. In order to illustrate the notion of apparent impact in our scenario, we consider a setting inwhich the trader starts waiting for a volume Q to accumulate on the ask queue at an initialtime t = 0 in which the system is characterized by coordinates ( V b , V a ), with V b < Q .As we are interested in characterizing the statistics of the price change x t and the hittingnumber n t as soon the volume at the ask queue reaches a volume Q , we define a stoppingtime T at which V a = Q at which the trader executes her order with a single trade.The main change with respect to the free case discussed above is the fact that the diffusionof the coordinates ( V b , V a ) no longer takes place on the positive orthant ( V b , V a ) ∈ (0 , ∞ ) × (0 , ∞ ), but on the semi-infinite strip ( V b , V a ) ∈ (0 , ∞ ) × (0 , Q ). We need correspondingly todefine a third type of probability associated with the absorption by the V a = Q boundary,which we denote as p ex ( t, V b , V a ). Its Laplace transform evolves according to equation (2)with the boundary conditions ˆ p ex ( ω, V b , Q ) = 1 − ˆ p ↑ ( ω, V b , Q ) = 1 − ˆ p ↓ ( ω, V b , Q ) = 1.Also in this case it is possible to show that ¯ p ↑ ( V b , V a ) + ¯ p ↓ ( V b , V a ) + ¯ p ex ( V b , V a ) = 1. Weremark that the bid-ask symmetry present in the free case is broken by the presence of the9 a = Q boundary, so that an asymmetric evolution of the price is expected in this setting.This is, in a nutshell, the reason why the apparent component of the impact emerges inour model.One can define the generating functions for the price x t and the hitting number n t whichare relevant for this problem by supposing that, as soon as the boundary V a = Q is reached,the price and the hitting number are frozen at their respective values at time t = T . Wedenote the relevant probabilities for this problem as P exx ( T, x ) and P exn ( T, x ), respectivelythe probability for the price and the hitting number to assume the value x and n when atthe stopping time T , and define their corresponding generating functions asΨ x ( ω, s ) = ∞ (cid:88) x = −∞ e − xs (cid:90) ∞ dT e − ωT P exx ( T, x ) (24)Ψ n ( ω, s ) = ∞ (cid:88) n = −∞ e − ns (cid:90) ∞ dT e − ωT P exn ( T, n ) . (25)These functions satisfy the set of equationsΨ x ( ω, s ) = ˆ p ex, + (cid:18) ˆ p ex, ↑ ˆ p ex, ↓ (cid:19) T ∞ (cid:88) n =0 (cid:18) ˆ p ↑↑ e − s ˆ p ↑↓ e − s ˆ p ↓↑ e s ˆ p ↓↓ e s (cid:19) n (cid:18) ˆ p ↑ e − s ˆ p ↓ e s (cid:19) (26)Ψ n ( ω, s ) = ˆ p ex, + (cid:18) ˆ p ex, ↑ ˆ p ex, ↓ (cid:19) T ∞ (cid:88) n =0 (cid:18) ˆ p ↑↑ ˆ p ↑↓ ˆ p ↓↑ ˆ p ↓↓ (cid:19) n (cid:18) ˆ p ↑ ˆ p ↓ (cid:19) e − ( n +1) s , (27)and as in the previous case can be solved by diagonalization of their respective transitionmatrices (App. B).Summarizing, in order to characterize analytically the price x T and the hitting number n T at the moment preceding the trade, we need to solve the diffusion problem (1) inthe modified geometry of a semi-infinite strip, and successively plug the probabilities ofhitting the boundaries into the generating functions Ψ x ( ω, s ) and Ψ n ( ω, s ), whose explicitexpressions have been worked out in App. B. The qualitative behavior of the price change x t at the time of execution t = T is similarin the µ = 0 and the µ > I ( Q ) = (cid:104) x T (cid:105) increases – in general non-monotonically –with the executed volume Q from the initial value (cid:104) x T (cid:105) = 0 corresponding to Q = V toan asymptotically constant value reached for Q → ∞ . The hitting number (cid:104) n T (cid:105) alwaysincreases monotonically in Q , growing unbounded from the value (cid:104) n T (cid:105) = 0 at Q = V up10o infinity. The precise form of the asymptotic scaling in Q depends crucially on µ : whilefor µ = 0 the growth of (cid:104) n T (cid:105) is asymptotically quadratic in Q , for µ > (cid:104) x T (cid:105) − (cid:104) x T (cid:105) , implying that apatient trader who is willing to execute a large order needs to be ready to face a potentiallylarge volatility risk. Notice that at large T one has (cid:104) n T (cid:105) ∼ ( (cid:104) x T (cid:105) − (cid:104) x T (cid:105) ) / (cid:29) (cid:104) x T (cid:105) ,due to the fact that the asymmetry induced in the diffusion problem by the absorbingboundary V a = Q is a smaller order effect with respect to the dominating symmetricbehavior associated with the Q → ∞ limit. Practically, this implies that the average valueof the apparent impact is at most of the order of one tick, while the variance of price isexpected to grow linearly in time as in the free case. We also remark that the evolutionof the momenta of x t and n t display a point of non-analyticity for Q = V lrg . This isexplained with the change of regime of the hitting probabilities for V lrg = Q : we have infact assumed that ¯ p ↑ = ¯ p ↓ = 1 − ¯ p ex = 0 independently of Q as long as Q < V lrg (in suchcase the trader can in fact execute all the volume instantly after a positive price change),while for
Q > V lrg we assume the hitting probabilities to satisfy Eq. (2).Fig. 2 show the result of a numerical simulation of the execution process, comparing itwith the semi-analytical results obtained by integrating numerically the expressions for theaverage slippage and the hitting number.The next part of this section will be devoted to a more detailed description the qualitativebehavior sketched above. In particular we will use the results proved in the appendices inorder to extract analytically the leading behavior of x T and n T at small and large Q .By exploiting the results of the 1 /Q expansion showed in App. A, we find in fact that forlarge executed volumes the apparent impact function tends to I ( Q ) = (cid:104) x T (cid:105) −−−−→ Q →∞ χ µ = 01 + χ µ > , (28)which indicates that for a market with short-time mean reversion of price ( χ <
1) the meanprice change is asymptotically smaller than (cid:104) x T (cid:105) = 1 / µ = 0) or (cid:104) x T (cid:105) = 1 (for µ > /Q (cid:104) n T (cid:105) −−−−→ Q →∞ Q πV sml V lrg for µ = 02 √ πµ (cid:18) Q √ µ (cid:19) / e µ ((1+ √ Q − V sml − V lrg ) V sml sinh (cid:0) √ µV lrg (cid:1) + V lrg sinh (cid:0) √ µV sml (cid:1) for µ > , (29)so that the total number of price changes asymptotically grows either as Q (in the case µ = 0) or with the exponential law Q / exp( µ (1 + √ Q ) (when µ > (cid:104) x T (cid:105) − (cid:104) x T (cid:105) −−−−→ Q →∞ χ (cid:104) n T (cid:105) , (30)which indicates that at leading order the asymptotic volatility of the free problem is notaffected by the conditioning effect induced by the execution.The small Q asymptotics of the process (provided V sml < V < V lrg ) leads to a linearbehavior in the vicinity of the point Q = V for all the observables. In particular can usethe relation ˆ p ↑↑ = ˆ p ↓↑ = 0 valid for Q < V lrg in order to show that (cid:104) x T (cid:105) −−−−→ Q → V ( Q − V ) (cid:18) δ ↑ − δ ↓ − ˆ p ↑↓ − ˆ p ↓↓ (cid:19) (31) (cid:104) n T (cid:105) −−−−→ Q → V ( Q − V ) (cid:18) δ ↑ + δ ↓ p ↑↓ − ˆ p ↓↓ (cid:19) (32) (cid:104) x T (cid:105) − (cid:104) x T (cid:105) −−−−→ Q → V ( Q − V ) (cid:18) δ ↑ + δ ↓ (1 + ˆ p ↓↓ )(1 − ˆ p ↑↓ )(1 − ˆ p ↓↓ ) (cid:19) , (33)where we have defined δ ↑ = V (cid:18) e π − e π + 1 (cid:19) for µ = 02 π (cid:90) ∞ dp β ( p ) p sin( pV ) p + µ (cid:18) e µV sinh( β ( p ) V ) (cid:19) for µ > δ ↓ = V (cid:18) π (cid:19) for µ = 0 µ + 2 π (cid:90) ∞ dp β ( p ) p sin( pV ) p + µ (cid:18) cosh( β ( p ) V ) − e µV sinh( β ( p ) V ) (cid:19) e µV for µ > , (35)with β ( p ) = (cid:112) p + 2 µ . Hence, the small volume behavior of the apparent impact isanalytical close to the starting volume V , in contrast with the ordinary price impact,which for small executed volumes grows roughly as Q η , being η an exponent between 0.6and 0.8 [4, 5, 6, 7, 8, 9, 10, 11]. 12 h x T i h n T i Q Execution time h T i µ = 1 µ = 1 / µ = 0 µ = 1 µ = 1 / µ = 0 µ = 1 µ = 1 / µ = 0Figure 2: Statistics for the one-shot execution problem. We represent the averages of theprice change (cid:104) x t (cid:105) , hitting number (cid:104) n t (cid:105) and execution time (cid:104) T (cid:105) as functions of the executedvolume Q , for different values of the drift µ . We have simulated 8000 realization of aprocess defined by the set of parameters V = 2 , V sml = 1 and V lrg = 3. The shadedregions correspond to the theoretical predictions, accounting for two-sigma regions, whilethe crosses are the results of numerical simulations.13 .2 Average execution time The execution time T follows a statistics similar to the one of the hitting number n T ,as showed in Fig. 2 where we display the evolution of the execution time as a functionof the executed volume V . As for the average price change and the hitting number, wehave compared with the simulations the semi-analytical result obtained by integratingnumerically Eqs. (58), (59) and (60) for the hitting probabilities and inserting them intothe Eqs. (98) and (99) for the momenta of T . We find that the average execution timeincreases monotonically from (cid:104) T (cid:105) = 0 for Q = V up to an asymptotic regime whose scalingdepends on the value of µ .In particular by performing a 1 /Q expansion of the execution time, we find the asymptoticscaling (cid:104) T (cid:105) −−−−→ Q →∞ (cid:104) n T (cid:105)(cid:104) t hit (cid:105) , (36)where we have used the results of App. A.4 in order to obtain the scaling of the hittingtimes (cid:104) t hit,α (cid:105) −−−−→ Q →∞ V b V a π log( Q ) for µ = 0 V b µ − π (cid:90) ∞ dp p sin( pV b )( p + µ ) e − ( √ p +2 µ − µ ) V a + µV b for µ > , (37)with ( V b , V a ) = ( V , V ) for α = 0 and ( V b , V a ) = ( V lrg , V sml ) for α ∈ {↑ , ↓} . In particular,the integral above is invariant under the exchange V b ↔ V a , so that (cid:104) t hit, ↑ (cid:105) , (cid:104) t hit, ↓ (cid:105) −−−−→ Q →∞ (cid:104) t hit (cid:105) (38)We remark that the Q → ∞ limit of (cid:104) t hit (cid:105) for µ > Q , so that the asymp-totic scaling of the average execution time is dominated by the exponential divergence ofthe hitting number. Also notice that, even though the time required to hit the V a = Q boundary is also diverging at a speed proportional to Q , its contribution is subleading:The barrier V a = Q needs to be hit just once, as opposed to the V a , V b = 0 boundarieswhich are hit a large number of times.The phenomenology of the case µ = 0 is extremely different: while the hitting numberscales like Q , the average (cid:104) t hit,α (cid:105) is proportional to log Q , implying that the average exe-cution time scales as Q log Q . Indeed, in both cases the increase of the expectation timewith the volume justifies the large volatility risk which a trader faces in the large Q regime.In limit of small executed volumes, the average execution time is dominated by (cid:104) T (cid:105) −−−−→ Q → V (cid:104) t hit, (cid:105) , (39)where t hit, can be expanded around Q = V by using the results of App. A.4, allowing toexpress the average execution time as an increasing function of the difference Q − V .14 Discussion and extensions
The model that we have presented deliberately simplifies the rich structure of a real orderbook, and does not take into account several of its microscopic features. We believe,motivated by the result of numerical simulations and the inspection of empirical data, thatthe features neglected in the current version of the model can be progressively reintroducedwithout modifying its main qualitative predictions. In particular, the essential feature onwhich our model relies is the fact that any asymmetry in liquidity (i.e., imbalances betweenbid and ask volumes) induces a corresponding asymmetry in the future direction of price,a well-known stylized fact of market microstructure summarized by the statement that“(efficient) price is where the volume is not” [18, 21]. This is why we believe our findingsto be robust.It would nevertheless be interesting to encode some more realistic features in the model,so to obtain predictions quantitatively more accurate. In particular:1. A stochastic volume after the depletion of the queues (as opposed to the deterministicvalues V sml and V lrg ) can be introduced with little expense. The only change requiredin our equations is the substitution p α ( t, V b , V a ) → (cid:90) dV b dV a p α ( t, V b , V a ) π ± ( V b , V a ) , (40)where π ± ( V b , V a ) refers to the distribution of the volumes after the queues reset.Hence, the analytical results presented above allow to solve the model even in thismore general setting.2. The dynamics of the best queues in an actual order book evolves discontinuouslythrough jumps in volume, consistently with the description adopted in [14, 15]. UsingPoissonian jumps of the queues allows to capture this feature, at the expense of amore involved analytical procedure required to obtain the first-passage probabilitieson a semi-infinite domain. While this modification can be substantial for manycontracts, when the tick size is sufficiently large the neglection of this effect becomesprogressively less relevant: for liquid stocks in equity markets, the typical durationbetween events at the best queue is of the order of the tens of milliseconds, whilewhen executing an order, the focus is typically on times larger then some seconds [22].3. The assumption of independence for the Fokker-Planck equations for the two queuesis a rather drastic simplification (dropped, for example, in [22]). Moreover, theempirical results of Ref. [19] indicate the actual Fokker-Planck equation describingthe diffusive limit of the queues dynamics has an even more complex structure thanthe one considered in [22]. Keeping this effect into account is likely to force one tocalculate the absorption probabilities numerically, due to the involved structure ofthe resulting Fokker-Planck equation. 15. The effect of a bid-ask asymmetry in the parameters of the model, as consideredin [22], can also be addressed, so to model for example the presence of local pricetrends.All these extensions modify the absorption probabilities p α,α (cid:48) while leaving invariant theexpressions for the generating functions Φ x ( ω, s ) , Φ( ω, s ) , Ψ x ( ω, s ) , Ψ n ( ω, s ), which can thenbe employed in order to study any Markovian model in which the queues are reset afterexhaustion. In particular, when closed-form expressions for the absorption probabilitiesare not available, it is possible to compute them via Monte Carlo. This allows to relate theempirical results concerning the time of exhaustion of the queues with the expected pricechange following a one-shot execution.In Fig. 3 we investigate how our results vary by introducing granularity both in time andvolume (we consider Poissonian jumps of discrete size), and by allowing the volumes of thequeues after reset to fluctuate.The key technical assumption of our approach is the one of Markovianity, which we needin order to be able to solve the model. While this assumption is justified for large tickcontracts [15], in order to model products of progressively smaller tick it would be necessaryto explore the spacial structure of the book by enlarging the number of price levels aroundthe best price included in the description. This work shows that even in a risk-neutral setting and in absence of a price trend, themarket impact of an order cannot be completely avoided by delaying the transaction:even if a one-shot strategy is adopted, thus completely removing the traditional notionof price impact from the cost of the execution strategy, such a cost reappears due to anasymmetric conditioning effect induced by the execution strategy. Moreover, the volatilityrisk associated with such a passive strategy can be (exponentially) large if the volume toexecute is too ingent. This indicates that the access to liquidity has a price which is to someextent unavoidable. Within our analysis we have characterized such cost, its fluctuations,and provided the probability of completely executing a one-shot order in a stylized marketmodel, relating these quantities to microstructural parameters. Even though we haveexpressed our ideas in a rather ad-hoc setting, we want to underline that even in a moregeneral scenario these qualitative predictions are expected to be observed: any type ofcorrelation among direction of price changes and liquidity may cause asymmetric effectswhich can induce an apparent impact effect for passive traders. In particular, as long asasymmetry in the volumes of the best queues are predictive for the direction of future pricechanges, we expect this effect to be detected. Notice also that the presence of opportunistic,one-shot traders would induce a liquidity-induced mean reversion effect in the price due to16 Q h x T i h x T i − h x T i Q dV = 0 dV = 0 . dV = 0 . dV = 1 Figure 3: Average price changes (cid:104) x T (cid:105) and price fluctuations (cid:104) x T (cid:105) − (cid:104) x T (cid:105) following theexecution of a trade of variable volume Q . The left plot shows the effect of granularityin time and volume, by considering the queue volumes to be subject to jumps of discretevolume at random Poissonian time. We fixed the parameters of the model in order tomatch µ = 0 and D = 1 in the continuous-volume limit of the model. As in Fig. 2, we havetaken V sml = 1 and V lrg = 3. We have used here a parameter dV interpolating among thecontinuous case dV = 0 and the rougher case dV = 1. The right plot superposes to thiseffect the one of a stochastic volume for the queues after reset ( V lrg uniformly distributedin { , . , , . , } and V sml uniformly distributed in { . , , . } ). The plots result from5000 realizations of the execution process, and suggest that the predictions of the modelare robust with respect to the effects described in Sec 4: starting from dV = 0 .
3, the curvesare almost superposed to the ones obtained in the continuous volume case.their tendency to exhaust large volume queues, which typically appear on the side of themarket which is opposite to the one where the price is currently trending.As a future research perspective, we look forward for the possibility of validating thequalitative predictions of this model through an inspection of proprietary data describingthe execution of one-shot orders.
Acknowledgements
The author acknowledges interesting discussions with J.-P. Bouchaud, N. Cosson andB. T´oth during the preparation of this manuscript. This research benefited from thesupport of the “Chair Markets in Transition”, under the aegis of “Louis Bachelier Finance17nd Sustainable Growth” laboratory, a joint initiative of ´Ecole Polytechnique, Universit´ed’´Evry Val d’Essonne and F´ed´eration Bancaire Fran¸caise.
A Diffusion in a semi-infinite strip
In this appendix we will be interested in computing the probabilities p ex ( t, V b , V a ), p ↑ ( t, V b , V a ), p ↓ ( t, V b , V a ) introduced in Sec. 2, denoting respectively the probability of executing a tradeof volume Q , emptying the ask queue and emptying the bid queue in a time t starting frombid and ask volumes respectively equal to V b and V a . We will focus exclusively on thecase in which a trader is present, reminding that free regime discussed in Sec. 2.2 can berecovered by taking the limit Q → ∞ , in which the execution probability p ex ( t, V b , V a )becomes zero. Finally, we remind that this problem corresponds to the one of finding thefirst-passage probabilities of a diffusing particle through any of the three boundaries of thesemi-infinite strip (0 , ∞ ) × (0 , Q ).Such first-passage probabilities ˆ p α satisfy the set of the independent equations12 ∇ ˆ p α ( ω, V ) − ( ∇ · µ ) ˆ p α ( ω, V ) · = ω ˆ p α ( ω, V ) (41)where ∇ = ( ∂ x , ∂ y ), µ = ( µ, µ ), V = ( V b , V a ) and α ∈ { ex, ↑ , ↓} . These equations differfor the boundary conditions as specified in Sec. 2. Specifically, p α ( ω, V ) is equal to zeroon all the boundaries of the region (0 , ∞ ) × (0 , Q ) except for the one corresponding to theevent labeled by α (i.e., ˆ p ex = 1 for V a = Q , ˆ p ↑ = 1 for V a = 0, ˆ p ↓ = 1 for V b = 0). Thechoice ω = 0 is associated to the probability of hitting a specific boundary in any point intime, in whose case the problem reduces to a simpler problem, for which we will be ableto provide explicit solutions.In order to solve this problem with elementary methods, we define the functions φ α ( ω, V ) =exp( − µ ( V a + V b ))ˆ p α ( ω, V ), which satisfy the simpler set of Helmholtz equations ∇ φ α ( ω, V ) = 2( ω + µ ) φ α (42)subject to the modified boundary conditions φ ex ( ω, V b , Q ) = e − µ ( V a + Q ) (43) φ ↑ ( ω, V b ,
0) = e − µV b (44) φ ↓ ( ω, , V a ) = e − µV a . (45)The problem of determining φ α is more conveniently handled by treating the semi-infinitestrip (0 , ∞ ) × (0 , Q ) as the P → ∞ limit of the rectangle (0 , P ) × (0 , Q ), in whose geometry18he general solution of the problem (41) can be written as follows: φ ex ( ω, V ) = − (cid:90) P dζ (cid:20) ∂∂η G ( V , ζ, η ) (cid:21) η = Q φ ex ( ω, ζ, Q ) (46) φ ↑ ( ω, V ) = (cid:90) P dζ (cid:20) ∂∂η G ( V , ζ, η ) (cid:21) η =0 φ ↑ ( ω, ζ,
0) (47) φ ↓ ( ω, V ) = (cid:90) Q dη (cid:20) ∂∂ζ G ( V , ζ, η ) (cid:21) ζ =0 φ ↓ ( ω, , η ) (48)where the Green function G ( V , ζ, η ) admits the two forms G ( V , ζ, η ) = 2 P ∞ (cid:88) n =1 sin( p n V b ) sin( p n ζ ) β n sinh( β n Q ) H n ( V a , η ) (49)= 2 Q ∞ (cid:88) n =1 sin( q n V b ) sin( q n ζ ) µ n sinh( µ n a ) Q n ( V b , ζ ) , (50)and where we have defined p n = πnP , β n = (cid:112) p n + 2 ω + 2 µ , (51) q n = πnQ , µ n = (cid:112) q n + 2 ω + 2 µ . (52)Finally, H n ( V a , η ) = (cid:26) sinh( β n η ) sinh( β n ( Q − V a )) if V a > η sinh( β n V a ) sinh( β n ( Q − η )) if η > V a (53) Q n ( V b , ζ ) = (cid:26) sinh( µ n ζ ) sinh( µ n ( P − V b )) if V b > ζ sinh( µ n V b ) sinh( µ n ( P − ζ )) if ζ > V b (54)The explicit form of the solution for the φ α can be written after performing the aboveintegrals and taking the P → ∞ limit. A further transformation back to the originalfunctions ˆ p α finally allows to express the solution as the seriesˆ p ex ( ω, V ) = 2 π (cid:88) n sin( q n V a ) n ( − n +1 e − µ ( Q − V a ) ω + µ ) Q /π n (1 − e − ( µ n − µ ) V b ) (55)ˆ p ↑ ( ω, V ) = 2 π (cid:88) n sin( q n V a ) n e µV a ω + µ ) Q /π n (1 − e − ( µ n − µ ) V b ) (56)ˆ p ↓ ( ω, V ) = 2 π (cid:88) n sin( q n V a ) n e µV a µ Q /π n (1 − ( − n e − µQ ) e − ( µ n − µ ) V b , (57)19r equivalently as the integralˆ p ex ( ω, V ) = 2 π (cid:90) ∞ dp pp + µ sin( pV b ) sinh( β ( p ) V a )sinh( β ( p ) Q ) e µV b − µ ( Q − V a ) (58)ˆ p ↑ ( ω, V ) = 2 π (cid:90) ∞ dp pp + µ sin( pV b ) sinh( β ( p )( Q − V a ))sinh( β ( p ) Q ) e µV b + µV a (59)ˆ p ↓ ( ω, V ) = 2 π (cid:90) ∞ dp pp + µ + 2 ω (cid:18) sin( pV b )sinh( β ( p ) Q ) (cid:19) e µV b (60) × (cid:18) sinh( β ( p ) Q ) − e − µ ( Q − V a ) sinh( β ( p ) V a ) − e µV a sinh( β ( p )( Q − V a )) (cid:19) . A.1 Hitting probabilities in the free, driftless case
In the special case µ = ω = 0 it is possible to sum analytically the above series so toexpress the result in term of elementary functions. It is sufficient to remind that ∞ (cid:88) n =1 ( − n +1 e zn n = log(1 + e zn ) (61)and to expand the trigonometric functions in term of exponentials in order to reduce theabove series to sums of logarithms. Then, by exploiting the identities i (cid:16) log(1 ± e iπV a /Q e − πV b /Q ) − log(1 ± e − iπV a /Q e − πV b /Q ) (cid:17) = arctan (cid:18) sin( πV a /Q ) e πV b /Q ± cos( πV a /Q ) (cid:19) (62)and 2 π ∞ (cid:88) n =1 sin( πV a /Q ) n = 1 − V a Q (63)one can obtain the following expression for the hitting probabilities:¯ p ex ( V ) = V a Q − π arctan (cid:18) sin( πV a /Q ) e πV b /Q + cos( πV a /Q ) (cid:19) (64)¯ p ↑ ( V ) = 1 − V a Q − π arctan (cid:18) sin( πV a /Q ) e πV b /Q − cos( πV a /Q ) (cid:19) (65)¯ p ↓ ( V ) = 2 π arctan (cid:18) sin( πV a /Q ) e πV b /Q − cos( πV a /Q ) (cid:19) + 2 π arctan (cid:18) sin( πV a /Q ) e πV b /Q + cos( πV a /Q ) (cid:19) . (66)Interestingly enough, the dependence on ω of the absorption probabilities close to the point ω = 0 is regular enough to lead to finite mean hitting times for any of the boundaries, as20pposed to the µ = 0 case of the free diffusion problem, in which all these quantities weredivergent. A.2 Small ω expansion in the free case The asymptotic analysis of Eqs. (58), (59) and (60) is particularly interesting, and hasbeen used in order to determine the large time behavior of the free diffusion problem(corresponding to the Q = ∞ , ω → µ : for µ > ω = 0, and thus can beexpanded as ˆ p α ( ω, V ) = ¯ p α ( V ) (cid:18) − ω (cid:104) t α ( V ) (cid:105) + 12 ω (cid:104) t α ( V ) (cid:105) + O ( ω ) (cid:19) (67)where the symbols (cid:104) t α ( V ) (cid:105) and (cid:104) t α ( V ) (cid:105) refer to the average hitting times conditional tothe initial condition V = ( V b , V a ) and a first absorption though a boundary of type α .An integral representation these terms can be obtained straightforwardly by differentiationwith respect to ω under the integral sign.For µ = 0 the hitting probabilities are non-analytic around ω = 0, and admit in particularthe expansion ˆ p α ( ω, V ) = (cid:88) n a ( n ) α ( V ) ω n + log( ω ) (cid:88) n b ( n ) α ( V ) ω n , (68)with a (0) α = ¯ p α (0) and b (0) α = 0. The subleading terms in the expansion are a (1) ( V ) = V b V a π (cid:18) (2 γ −
3) + 2 V a V b arctan (cid:18) V b V a (cid:19) + log (cid:18) ( V b ) + ( V a ) (cid:19)(cid:19) (69) b (1) ( V ) = V b V a π , (70)where γ denotes the Euler-Mascheroni constant. These terms can be obtained by using theseries representation for the absorption probabilitiesˆ p ↑ ( ω, V ) = 2 π ∞ (cid:88) n =0 ( − n (2 n + 1)! (cid:18) V b V a (cid:19) n +1 (cid:90) ∞ z n e − √ z +2 ω ( V a ) . (71)The correction terms (69) and (70) are finally recovered after exploiting the identity (cid:82) ∞ z n e −√ z + α = (2 n − α n +1 K n +1 ( α ), with K n +1 ( α ) denoting the modified Besselfunction of the second type of order n + 1 calculated in α . Notice in particular that thedivergence of the Bessel function close to α = 0 is canceled by the factor α n +1 .21 .3 Hitting probabilities in the large Q regime The regime of large Q and ω = 0 has been analyzed in order to find the asymptotic scalingof the price change x t and of the hitting number n t in the execution problem illustrated inSec. 3. In the case µ >
0, Eqs. (58), (59) and (60) can be expanded as¯ p ex ( V , Q ) −−−−→ Q →∞ ¯ p ex ( V , ∞ ) + x √ πµ (cid:32) √ µQ (cid:33) / sinh (cid:16) √ µy (cid:17) e µ ( x + y − (1+ √ Q ) (72)¯ p ↑ ( V , Q ) −−−−→ Q →∞ ¯ p ↑ ( V , ∞ ) − x √ πµ (cid:32) √ µQ (cid:33) / sinh (cid:16) √ µy (cid:17) e µ ( x + y − √ Q ) (73)¯ p ↓ ( V , Q ) −−−−→ Q →∞ ¯ p ↓ ( V , ∞ ) − x √ πµ (cid:32) √ µQ (cid:33) / sinh (cid:16) √ µy (cid:17) e µ ( x + y − (1+ √ Q ) , (74)while for µ = 0 one can simply differentiate Eqs. (64), (65) and (66) and obtain¯ p ex ( V , Q ) −−−−→ Q →∞ ¯ p ex ( V , ∞ ) + πV a V b Q (75)¯ p ↑ ( V , Q ) −−−−→ Q →∞ ¯ p ↑ ( V , ∞ ) − πV a V b Q (76)¯ p ↓ ( V , Q ) −−−−→ Q →∞ ¯ p ↓ ( V , ∞ ) − πV a V b Q . (77)Above expressions can be inserted in the formulae for (cid:104) x T (cid:105) and (cid:104) n T (cid:105) calculated in App. Bso to estimate their large Q scaling. A.4 Unconditional hitting time
The unconditional hitting time t hit ( V ), measuring the average time required to hit any ofthe boundaries, is defined in App.B, where its averages are expressed as (cid:104) t hit ( V ) (cid:105) = − ddω (ˆ p ex + ˆ p ↑ + ˆ p ↓ ) (cid:12)(cid:12)(cid:12)(cid:12) ω =0 , (78)We want to show that it can be conveniently computed by first summing Eqs. (58), (59)and (60), and successively performing the derivative with respect to ω , in order to exploitthe cancellations among the summands. We obtain in particular (cid:104) t hit ( V ) (cid:105) = 4 π (cid:90) ∞ dp p ( p + µ ) sin( pV b ) (79) × (cid:18) e µV b − sinh( β ( p ) V a )sinh( β ( p ) Q ) e µ ( V b + V a − Q ) − sinh( β ( p )( Q − V a ))sinh( β ( p ) Q ) e µ ( V b + V a ) (cid:19) , µ = 0 reduces to (cid:104) t hit ( V ) (cid:105) = 4 Q π (cid:90) ∞ dp sin( pV b /Q ) p [1 − cosh( pV a /Q ) + tanh( p/
2) sinh( pV a /Q )] . (80) B Solution of a Markovian market model
B.1 Generating functions for the free case
In Sec. 2 we have considered the problem of calculating the quantities P x ( t, x ) and P n ( t, n )describing the probability for the price and the hitting number to take values respectivelyof x and n at time t . The former one can be written as P x ( t, x ) = ∞ (cid:88) n =0 (cid:88) α · · · (cid:88) α n +1 (cid:90) ∞ (cid:32) n +1 (cid:89) i =0 dt i (cid:33) δ (cid:32) t − n +1 (cid:88) i =0 t i (cid:33) δ (cid:32) x − n +1 (cid:88) i =1 X ( α i ) (cid:33) × (cid:18)(cid:90) t n +1 dt (cid:48) (1 − p ↑ α n +1 ( t (cid:48) ) − p ↓ α n +1 ( t (cid:48) )) (cid:19) (cid:32) n (cid:89) i =1 p α i +1 α i ( t i ) (cid:33) p α + (cid:90) t dt (1 − p ↑ ( t ) − p ↓ ( t )) δ ( x ) , (81)where X ( ↑ ) = − X ( ↓ ) = 1. Each of the terms in above equation can be representedschematically as in Fig. 4, in which we show that the n -th term of the sum comprises n + 1 absorptions from the starting time to the last one t , while the succession of indexes { α i } n +1 i =1 ∈ {↑ , ↓} n +1 labels the type of boundary hit during absorption number i . Theprobability P n ( n, t ) can be obtained from the previous expression after the substitution X ( α i ) →
1. The generating function associated to the probability distribution (81) isreadily found by integrating in the t and x coordinates, and can be expressed compactlyin matrix form as in Eq. (3). The generating function for the hitting number is foundanalogously, leading to Eq. (4). We can rewrite Eqs. (3) and (4) more succinctly asΦ x ( ω, s ) = 1 ω (cid:34) ˆ p nohit + ( ˆ p nohit ) T ∞ (cid:88) n =0 ( ˆ P hit K x ) n K x ˆ p hit (cid:35) (82)Φ n ( ω, s ) = 1 ω (cid:34) ˆ p nohit + ( ˆ p nohit ) T ∞ (cid:88) n =0 ( ˆ P hit K n ) n K n ˆ p hit (cid:35) , (83)where we have defined the matricesˆ P hit = (cid:18) ˆ p ↑↑ ˆ p ↑↓ ˆ p ↓↑ ˆ p ↓↓ (cid:19) K x = (cid:18) e − s e s (cid:19) K n = (cid:18) e − s e − s (cid:19) , (84)23 = 0 tt t t n t n +1 ↵ ↵ ↵ n ↵ n +1 t ↵ . . .. . .p ↵ , p ↵ ,↵ p ↵ ,↵ p ↵ n +1 ,↵ n Z p nohit↵ n +1 | {z } evolution operator P hit Figure 4: Schematic representation of a generic term of the evolution equation (81) for theproblem of the free evolution of the price.together with the vectorsˆ p hit = (cid:18) ˆ p ↑ ˆ p ↓ (cid:19) ˆ p nohit = (cid:18) − ˆ p ↑↑ − ˆ p ↓↑ − ˆ p ↑↓ − ˆ p ↓↓ (cid:19) (85)and the scalar ˆ p nohit = 1 − ˆ p ↑ − ˆ p ↓ . Notice the factor ω − appearing in Eqs. (82) and (83),which arises from the integration by parts of the last time t n +1 . The infinite sum appearingin the expression for the generating functions can be performed explicitly by passing toprincipal component. One obtainsΦ x ( ω, s ) = 1 ω (cid:104) ˆ p nohit + ( ˆ p nohit ) T U x ( I − Λ x ) − U − x ˆ p hit (cid:105) (86)Φ n ( ω, s ) = 1 ω (cid:104) ˆ p nohit + ( ˆ p nohit ) T U n ( I − Λ n ) − U − n ˆ p hit (cid:105) , (87)where I denotes the identity matrix, and where we have introduced the eigenvalue decom-positions ˆ P hit K x = U x Λ x U − x and ˆ P hit K n = U n Λ n U − n . The matrix product appearing inthe first of above equations results U x ( I − Λ x ) − U − x = 11 − ˆ p ↑↓ ˆ p ↓↑ + ˆ p ↑↑ ˆ p ↓↓ − e − s ˆ p ↑↑ − e s ˆ p ↓↓ (cid:18) − e s ˆ p ↓↓ e − s ˆ p ↑↓ e s ˆ p ↓↑ − e − s ˆ p ↑↑ (cid:19) , (88)while the one relative to the second equation can be obtained through the substitutionˆ p ↓ α → e − s ˆ p ↓ α .The differentiation of the generating functions with respect to s leads finally to the Laplacetransforms of mean price change (cid:104) ˆ x ω (cid:105) and hitting number (cid:104) ˆ n ω (cid:105) , together with their squares24 ˆ x ω (cid:105) and (cid:104) ˆ n ω (cid:105) : (cid:104) ˆ x ω (cid:105) = 1 ω (cid:18) ˆ p ↑ (1 − ˆ p ↓↓ − ˆ p ↓↑ ) − ˆ p ↓ (1 − ˆ p ↑↑ − ˆ p ↑↓ )1 − ˆ p ↑↓ ˆ p ↓↑ + ˆ p ↑↑ ˆ p ↓↓ − ˆ p ↑↑ − ˆ p ↓↓ (cid:19) (89) (cid:104) ˆ x ω (cid:105) = ˆ p ↑ ω (cid:34) (1 − p ↓↓ + ˆ p ↓↓ − ˆ p ↓↑ + 3ˆ p ↓↓ ˆ p ↓↑ − ˆ p ↓↑ ˆ p ↑↓ − ˆ p ↓↓ ˆ p ↓↑ ˆ p ↑↓ )(1 − ˆ p ↓↓ − ˆ p ↑↑ − ˆ p ↑↓ ˆ p ↓↑ + ˆ p ↑↑ ˆ p ↓↓ ) (90)+ ˆ p ↓↑ ˆ p ↑↓ + ˆ p ↑↑ − p ↓↓ ˆ p ↑↑ + ˆ p ↓↓ ˆ p ↑↑ − ˆ p ↓↑ ˆ p ↑↑ − ˆ p ↓↓ ˆ p ↓↑ ˆ p ↑↑ (1 − ˆ p ↓↓ − ˆ p ↑↑ − ˆ p ↑↓ ˆ p ↓↑ + ˆ p ↑↑ ˆ p ↓↓ ) (cid:35) + ( ↑ exch. ↓ ) (cid:104) ˆ n ω (cid:105) = 1 ω (cid:18) ˆ p ↑ (1 − ˆ p ↑↑ + ˆ p ↓↑ ) + ˆ p ↓ (1 − ˆ p ↑↑ + ˆ p ↑↓ )1 − ˆ p ↑↓ ˆ p ↓↑ + ˆ p ↑↑ ˆ p ↓↓ − ˆ p ↑↑ − ˆ p ↓↓ (cid:19) (91) (cid:104) ˆ n ω (cid:105) = ˆ p ↑ ω (cid:34) (1 − p ↓↓ + ˆ p ↓↓ + 3ˆ p ↓↑ − ˆ p ↓↓ ˆ p ↓↑ + 3ˆ p ↓↑ ˆ p ↑↓ − ˆ p ↓↓ ˆ p ↓↑ ˆ p ↑↓ )(1 − ˆ p ↓↓ − ˆ p ↑↑ − ˆ p ↑↓ ˆ p ↓↑ + ˆ p ↑↑ ˆ p ↓↓ ) (92)+ ˆ p ↓↑ ˆ p ↑↓ + ˆ p ↑↑ − p ↓↓ ˆ p ↑↑ + ˆ p ↓↓ ˆ p ↑↑ − ˆ p ↓↑ ˆ p ↑↑ − ˆ p ↓↓ ˆ p ↓↑ ˆ p ↑↑ (1 − ˆ p ↓↓ − ˆ p ↑↑ − ˆ p ↑↓ ˆ p ↓↑ + ˆ p ↑↑ ˆ p ↓↓ ) (cid:35) + ( ↑ exch. ↓ ) . B.2 Generating functions for the one-shot execution problem
The problem which we are required to solve during the execution of a one-shot orderrequires finding the generating functions for the probabilities P exx ( T, x ) and P exn ( T, x ) thatthe price and the hitting number assume respectively the values of x and n at the stoppingtime T . Those probabilities follow a law extremely similar to Eq. (81), we have in fact P exx ( t, x ) = ∞ (cid:88) n =0 (cid:88) α · · · (cid:88) α n +1 (cid:90) ∞ (cid:32) n +1 (cid:89) i =0 dt i (cid:33) δ (cid:32) t − n +1 (cid:88) i =0 t i (cid:33) δ (cid:32) x − n +1 (cid:88) i =1 X ( α i ) (cid:33) × p ex,α n +1 ( t n +1 ) (cid:32) n (cid:89) i =1 p α i +1 α i ( t i ) (cid:33) p α + p ex, ( t ) δ ( x ) , while an analogous expression holds for P exn ( T, n ). In particular, the above equation canbe recovered from Eq. (81) by exploiting the different terminal condition (cid:90) t dt (cid:48) (1 − p ↑ α ( t (cid:48) ) − p ↓ α ( t (cid:48) )) → p ex,α ( t ) , (93)which in Laplace space reads1 ω [1 − ˆ p ↑ α ( ω ) − ˆ p ↓ α ( ω )] → ˆ p ex,α ( ω ) . (94)25he generating functions for this modified problem can then be found as shown in theprevious section by taking into account the substitution (94).We are interested in particular in calculating the statistics of the price and the hittingnumber at the stopping time T , which can recovered by differentiation with respect to s of the generating function evaluated at the point ω = 0. We notice in particular thatby substituting ˆ p α,α (cid:48) → ¯ p α,α (cid:48) , and by exploiting the relation ¯ p ex,α = 1 − ¯ p ↑ ,α − ¯ p ↓ ,α inEqs. (89), (90), (91) and (92), it is possible to use these same formulae even for the executionproblem by simply multiplying by ω , thus removing the overall ω − factor.Finally, we would like to obtain the statistics of the stopping time T itself, which can beobtained from the ω = 0 value of the generating functions Ψ( ω ) = Ψ x ( ω,
0) = Ψ n ( ω, ω ) = ˆ p ex, + ˆ p ↑ (ˆ p ↓↓ ˆ p ex, ↑ − ˆ p ↓↑ ˆ p ex, ↓ − ˆ p ex, ↑ ) + ˆ p ↓ (ˆ p ↑↑ ˆ p ex, ↓ − ˆ p ↑↓ ˆ p ex, ↑ − ˆ p ex, ↓ )(1 − ˆ p ↓↓ − ˆ p ↑↑ − ˆ p ↑↓ ˆ p ↓↑ + ˆ p ↑↑ ˆ p ↓↓ ) . (95)In the hypothesis in which those functions are at least q times differentiable at ω = 0 (validfor µ >
0) one can write ∂ qω ˆ p α,α (cid:48) | ω =0 = ( − q ¯ p α,α (cid:48) (cid:104) t qα,α (cid:48) (cid:105) . (96)The unconditional hitting time t hit,α (cid:48) is defined as the time required for any of the bound-aries to be hit. Its moment of order q is related to the conditional ones through therelation (cid:104) t qhit,α (cid:48) (cid:105) = (cid:88) α ¯ p α,α (cid:48) (cid:104) t qα,α (cid:48) (cid:105) . (97)By employing these definitions, one can prove that (cid:104) T (cid:105) = 1 B ( A ↓ (cid:104) t hit, ↓ (cid:105) + A ↑ (cid:104) t hit, ↑ (cid:105) + B (cid:104) t hit, (cid:105) ) , (98)and (cid:104) T (cid:105) − (cid:104) T (cid:105) = 1 B (cid:18) A ↓ [ (cid:104) t hit, ↓ (cid:105) − (cid:104) t hit, ↓ (cid:105) ] + B (cid:104) t hit, (cid:105) − (cid:104) t hit, (cid:105) ] (cid:19) (99)+ 1 B (cid:18) − ( A ↓ (cid:104) t hit, ↓ (cid:105) + A ↑ (cid:104) t hit, ↑ (cid:105) ) A ↓ B ( (cid:104) t hit, ↓ (cid:105) − (cid:104) t hit, (cid:105) ) − A ↓ B (cid:104) t hit, (cid:105) (cid:19) + 2 B (cid:18) A ↓ ¯ p ↑↓ (cid:104) t ↑↓ (cid:105) + A ↑ ¯ p ↑↑ (cid:104) t ↑↑ (cid:105) + B ˆ p ↑ (cid:104) t ↑ (cid:105) (cid:19)(cid:18) (1 − ¯ p ↓↓ ) (cid:104) t hit, ↑ (cid:105) + ¯ p ↓↑ (cid:104) t hit, ↓ (cid:105) (cid:19) + ( ↑ exch. ↓ ) , where we have defined A α = ¯ p − α ¯ p α, − α + ¯ p α − ¯ p α ¯ p − α, − α (100) B = 1 − ¯ p ↓↓ − ¯ p ↑↑ − ¯ p ↓↑ ¯ p ↑↓ + ¯ p ↑↑ ¯ p ↓↓ , (101)and introduced the convention ± ( ↑ ) = ∓ ( ↓ ).26 eferences [1] Jean-Philippe Bouchaud. Price impact. Encyclopedia of quantitative finance , 2010.[2] Jean-Philippe Bouchaud, J Doyne Farmer, and Fabrizio Lillo. How markets slowlydigest changes in supply and demand.
Handbook of financial markets: dynamics andevolution , 1:57, 2009.[3] Dimitris Bertsimas and Andrew W Lo. Optimal control of execution costs.
Journalof Financial Markets , 1(1):1–50, 1998.[4] N Torre. Barra market impact model handbook.
BARRA Inc., Berkeley , 1997.[5] Robert Almgren, Chee Thum, Emmanuel Hauptmann, and Hong Li. Direct estimationof equity market impact.
Risk , 18:5752, 2005.[6] Esteban Moro, Javier Vicente, Luis G Moyano, Austin Gerig, J Doyne Farmer,Gabriella Vaglica, Fabrizio Lillo, and Rosario N Mantegna. Market impact and tradingprofile of hidden orders in stock markets.
Physical Review E , 80(6):066102, 2009.[7] Bence Toth, Yves Lemperiere, Cyril Deremble, Joachim De Lataillade, Julien Kock-elkoren, and J-P Bouchaud. Anomalous price impact and the critical nature of liquidityin financial markets.
Physical Review X , 1(2):021006, 2011.[8] Albert Pete Kyle and Anna Obizhaeva. Large bets and stock market crashes.
Availableat SSRN 2023776 , 2012.[9] Nataliya Bershova and Dmitry Rakhlin. The non-linear market impact of large trades:evidence from buy-side order flow.
Quantitative Finance , 13(11):1759–1778, 2013.[10] C Gomes and H Waelbroeck. Is market impact a measure of the information value oftrades? market response to liquidity vs. informed metaorders.
Quantitative Finance ,15(5):773–793, 2015.[11] Iacopo Mastromatteo, Bence T´oth, and Jean-Philippe Bouchaud. Agent-based modelsfor latent liquidity and concave price impact.
Phys. Rev. E , 89:042805, Apr 2014.[12] Emmanuel Bacry, Adrian Iuga, Matthieu Lasnier, and Charles-Albert Lehalle. Marketimpacts and the life cycle of investors orders.
Available at SSRN 2532152 , 2014.[13] Elia Zarinelli, Michele Treccani, J Doyne Farmer, and Fabrizio Lillo. Beyond thesquare root: Evidence for logarithmic dependence of market impact on size and par-ticipation rate. arXiv preprint arXiv:1412.2152 , 2014.[14] Rama Cont, Sasha Stoikov, and Rishi Talreja. A stochastic model for order bookdynamics.
Operations research , 58(3):549–563, 2010.2715] Rama Cont and Adrien De Larrard. Price dynamics in a markovian limit order market.
SIAM Journal on Financial Mathematics , 4(1):1–25, 2013.[16] Bruno Biais, Pierre Hillion, and Chester Spatt. An empirical analysis of the limit orderbook and the order flow in the paris bourse. the Journal of Finance , 50(5):1655–1689,1995.[17] Zoltan Eisler, Jean-Philippe Bouchaud, and Julien Kockelkoren. The price impactof order book events: market orders, limit orders and cancellations.
QuantitativeFinance , 12(9):1395–1419, 2012.[18] Rama Cont, Arseniy Kukanov, and Sasha Stoikov. The price impact of order bookevents.
Journal of financial econometrics , 12(1):47–88, 2013.[19] A Gareche, G Disdier, J Kockelkoren, and J-P Bouchaud. Fokker-planck descriptionfor the queue dynamics of large tick stocks.
Physical Review E , 88(3):032809, 2013.[20] Michael J Kearney and Satya N Majumdar. On the area under a continuous timebrownian motion till its first-passage time.
Journal of Physics A: Mathematical andGeneral , 38(19):4097, 2005.[21] Sylvain Delattre, Christian Y Robert, and Mathieu Rosenbaum. Estimating the effi-cient price from the order flow: a brownian cox process approach.
Stochastic Processesand their Applications , 123(7):2603–2619, 2013.[22] Rama Cont and Adrien De Larrard. Order book dynamics in liquid markets: limittheorems and diffusion approximations. arXiv preprint arXiv:1202.6412arXiv preprint arXiv:1202.6412