optimal execution and price manipulations in time-varying limit order books

8/9/2019 Optimal Execution and Price Manipulations in Time-Varying Limit Order Books

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arXiv:1204.2736v1

[q-fin.TR]

12Apr2012

Optimal execution and price manipulations in time-varying limit

order books

Aurlien Alfonsi and Jos Infante Acevedo

Universit Paris-Est, CERMICS,

Project team MathRISK ENPC-INRIA-UMLV

Ecole des Ponts, 6-8 avenue Blaise Pascal, 77455 Marne-la-valle, France

[email protected], [email protected]

April 13, 2012

Abstract: This paper focuses on an extension of the Limit Order Book (LOB) model with general shapeintroduced by Alfonsi, Fruth and Schied [2]. Here, the additional feature allows a time-varying LOB depth.We solve the optimal execution problem in this framework for both discrete and continuous time strategies.This gives in particular sufficient conditions to exclude Price Manipulations in the sense of Huberman andStanzl [12]or Transaction-Triggered Price Manipulations (see Alfonsi, Schied and Slynko[4]). These condi-tions give interesting qualitative insights on how market makers may create or not price manipulations.

Key words: Market impact model, optimal order execution, limit order book, market makers, price ma-nipulation, transaction-triggered price manipulation.

AMS Class 2010: 91G99, 91B24, 91B26, 49K99

Acknowledgements: This work has benefited from the support of the Eurostars E!5144-TFM projectand of the Chaire Risques Financiers of Fondation du Risque. Jos Infante Acevedo is grateful to AXA

Resarch Fund for his doctoral fellowship.

Introduction

It is a rather standard assumption in finance to consider an infinite liquidity. By infinite liquidity, we meanhere that the asset price is given by a a single value, and that one can buy or sell any quantity at this pricewithout changing the asset price. This assumption is in particular made in the Black and Scholes model [7],and is often made as far as derivative pricing is concerned. When considering portfolio over a large timehorizon, this approximation is relevant since one may split orders in small ones along the time and reducesones own impact on the price. At most, the lack of liquidity can be seen as an additional transaction cost.This issue has been broadly investigated in the literature, see Cetin, Jarrow and Protter [8] and referenceswithin.

If we consider instead brokers that have to trade huge volumes over a short time period (some hoursor some days), we can no longer neglect the price impact of trading strategies. We have to focus on themarket microstructure and model how prices are modified when buy and sell orders are executed. Generallyspeaking, the quotation of an asset is made through a Limit Order Book (LOB) that lists all the waitingbuy and sell orders on this asset. The order prices have to be a multiple of the tick size, and orders at thesame price are arranged in a First-In-First-Out stack. The bid (resp. ask) price is the price of the highestwaiting buy (resp. lowest selling buy) order. Then, it is possible to buy or sell the asset in two differentways: one can either put a limit order and wait that this order matches another one or put a market order

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that consumes the cheapest limit orders in the book. In the first way, the transaction cost is known but theexecution time is uncertain. In the second way, the execution is immediate (provided that the book containsenough orders). The price per share instead depends on the order size. For a buy (resp. sell) order, the firstshare will be traded at the ask (resp. bid) price while the last one will be traded some ticks upper (resp.lower) in order to fill the order size. The ask (resp. bid) price is then modified accordingly.

The typical issue on a short time scale is the optimal execution problem: on given a time horizon, how to

buy or sell optimally a given amount of assets? As pointed in Gatheral [ 10] and Alfonsi, Schied and Slynko[4],this problem is closely related to the market viability and to the existence of price manipulations. Modellingthe full LOB dynamics is not a trivial issue, especially if one wants to keep tractability to solve then theoptimal execution problem. Instead, simpler models called market impact models have been proposed. Thesemodels only describe the dynamics of one asset price and model how the asset price is modified by a tradingstrategy. Thus, Bertsimas and Lo [6], Almgren and Chriss [5], Obizhaeva and Wang [13] have proposeddifferent models where the price impact is proportional to the trading size, in which they solve the optimalexecution problem. However, some empirical evidence on the markets show that the price impact of a tradeis not proportional to its size, but is rather proportional to a power of its size (see for example Potters andBouchaud[14], and references within). With this motivation in mind, Gatheral [10] has suggested a nonlinearprice impact model. In the same direction, Alfonsi, Fruth and Schied [2]have derived a price impact modelfrom a simple LOB modelling. Basically, the LOB is modelled by a shape function that describes the densityof limit orders at a given price. This model has then been studied further by Alfonsi and Schied [3] and

Predoiu, Shaikhet and Shreve[15].The present paper extends this model by letting the LOB shape function vary along the time. Beyond

solving the optimal execution problem in a more general context, our goal is to understand how the dynamicsof the LOB may create or not price manipulations. Indeed, a striking result in [2, 3] is that the optimalexecution strategy is made with trades of same sign, which excludes any price manipulation. This resultholds under rather general assumptions on the LOB shape function, when the LOB shape does not changealong the time. Instead, we will see in this paper that a time-varying LOB may induce price manipulationsand we will derive sufficient conditions to exclude them. These conditions are not only interesting froma theoretical point of view. They give a qualitative understanding on how price manipulations may occurwhen posting or cancelling limit orders. While preparing this work, Fruth, Schneborn and Urusov [9]havepresented a paper where this issue is addressed for a block-shaped LOB, which amounts to a proportionalprice impact. Here, we get back their result and extend them to general LOB shapes and thus nonlinearprice impact. The other contribution of this paper is that we solve the optimal execution in a continuous

time setting while [2, 3] mainly focus on discrete time strategies. This is in particular much more suitableto state the conditions that exclude price manipulations.

1 Market model and the optimal execution problem

1.1 The model description

The problem that we study in this paper is the classical optimal execution problem. To deal with thisproblem, we consider in this paper a framework which is a natural extension of the model proposed inAlfonsi, Fruth and Schied[2] and developed by Alfonsi and Schied[3] and Predoiu, Shaikhet and Shreve[15].The additional feature that we introduce here is to allow a time varying depth of the order book. We considera large trader who wants to liquidate a portfolio of shares in a time period of [0, T]. In order liquidate

these

shares, the large trader uses only market orders, that is buy or sell orders that are immediatelyexecuted at the best available current price. Thus, our large trader cannot put limit orders. A long position

> 0 will correspond to a sell program while a short position < 0 will stand for a a buy strategy. Theoptimal execution problem consists in finding the optimal trading strategy that minimizes the expected costof the large trader.

We assume that the price process without the large trader would be given by a rightcontinuous martingale(S0t , t 0) on a given filtered probability space (, (Ft), F,P). The actual price process (St, t 0) that

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takes into account the trades of the large trader is defined by:

St = S0t +Dt, t0. (1)

Thus, the process (Dt, t0) describes the price impact of the large trader. We also introduce the process(Et, t0) that describes the volume impact of the large trader. If the large trader puts a market order ofsizet (t > 0 is a buy order and t < 0 a sell order), the volume impact process changes from Et to:

Et+ := Et+t. (2)

When the large trader is not active, its volume impact Et goes back to 0. We assume that it decaysexponentially with a deterministic time-dependent rate t > 0 called resilience, so that we have:

dEt =tEtdt. (3)We now have to specify how the processes D and Eare related. To do so, we suppose a continuous

distribution buy and sell limit orders around the unaffected price S0t: forxR, we assume that the numberof limit orders available between prices S0t +x and S

0t +x+dx is given by (t)f(x)dx. These orders are

sell orders ifx Dt and buy orders otherwise. The functionsf : R (0, ) and : [0, T] (0, ) areassumed to be continuous, and represent respectively the LOB shape and the depth of the order book. Wedefine the antiderivative of the function f, F(y) := y

0 f(x)dx, y

R, and assume that

limx

F(x) =and limx

F(x) =, (4)

which means that the book contains an infinite number of limit buy and sell orders. Thus, we set thefollowing relationship between the volume impact Et and the price impact Dt: Dt

0

(t)f(x)dx= Et,

or equivalently,

Et = (t)F(Dt) and Dt = F1

Et(t)

. (5)

Within this framework, a large tradet changes Dt toDt+= F1 Et+t

(t)

and has the cost Dt+Dt

(S0t +x)(t)f(x)dx= tS0t +

Dt+Dt

(t)xf(x)dx:= t(t). (6)

Throughout the paper, we assume that isC2 and sett = (t)(t). Thus, we have

(t) =(0)exp

t0

udu

,

andtt isC1. Similarly, we assume that tt isC1.Now, let us observe that we have assumed that the volume impact decays exponentially when the large

trader is inactive. Other choices are of course possible, and a natural one would be to assume that the price

impact decays exponentiallydDt =tDtdt, (7)

which amounts to assume that dEt = tEtdt t(t)f(F1(Et/(t)))F1(Et/(t))dt by (5).Definition 1.1. The dynamics of modelV with volume impact reversion is the one given by (1), (2), (3)and (5). The dynamics of modelP with price impact reversion is the one given by (1), (2), (7) and (5).In both models, we assume that the market is at equilibrium at time0, i.e. E0 = D0 = 0.

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Remark 1.1. Though being simplistic, this model describes throught and(t) the two different ways thatmarket makers have to put (or cancel) limit orders: it is either possible to pile orders at an existing price orto put orders at a better price than the existing ones. Thus,(t) describes how market makers pile orderswhile t describes the rate at which new orders appear at a better price. Basically, one may think these

functions one-day periodic, with relative high values at the opening and the closing of the market and lowvalues around noon. The particular case 1 corresponds to the model introduced by Alfonsi, Fruth andSchied [2]for which new orders can only appear at a better price.

1.2 The optimal execution problem, and price manipulation strategies

We focus on the optimal liquidation of a portfolio with shares by a large trader who can place marketorders over a period of time [0, T]. Thus, > 0 (resp. < 0) corresponds to a selling (resp. buying)strategy.

We first consider discrete strategies and assume that at most N+ 1 trades can occur. An admissiblestrategy will be then described by an increasing sequence 0 = 0 N = T of stopping times andrandom variables0, . . . , N (i stands for the trading size at time i) such that

+

Ni=0i = 0, i.e. the trader liquidates indeed shares,

i is

Fi-measurable,

M R, 0iN, iM, a.s.The expected cost of an admissible strategy(, T)with = (0, . . . , N)andT = (0, . . . , N)is given by

C(, T) = E Ni=0

i(i)

, (8)

wherei(i) stands for the cost of the i-th trade, and is defined by (6) in models V or P. The goal of thelarge trader is then to minimize this expected cost among the admissible strategies.

We also consider continuous time trading strategy and make the same assumptions as Gatheral et al. [11].An admissible strategy (Xt)t0 is a stochastic process such that

X0 = andXT+= 0,

X is(Ft)-adapted and leftcontinuous, the function t[0, T+] Xt has finite and a.s. bounded total variation.

The processXt describes the number of shares that remains to liquidate at time t. Thus, the discrete timestrategy above corresponds toXt = +

Ni=0i1i


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which coincides with (8) for discrete strategies. Here,Xt = Xt+ Xt denotes the jump ofXat time t(jumps are countable), and dXt stands for the signed measure on [0, T] associated to (Xt, 0 t T+) (a

jumpXT induces a Dirac mass in T). If we introduce the continuous part ofX,Xct :=Xt

0s 0 shares is made only withintermediate buy trades and has thus a positive cost. Thus, by some cost continuity that usually holds (thisis the case for models V andP), any round trip has a nonnegative cost.

Remark 1.2. It is possible to define a two-sided limit order book model like in Alfonsi, Fruth and Schied[2]or Alfonsi and Schied ([3], Section 2.6). In such a model, bid and ask prices evolve as follows. A buy (resp.

sell) order of the large trader shifts the ask (resp. bid) price and leaves the bid (resp. ask) price unchanged.When the large trader is idle, the shifts on the ask and bid prices goes back exponentially to zero, like inmodels V or P. As in [2, 3], the two-sided model coincides with the model presented here when the largetrader puts only buy orders or only sell orders. In particular, the optimal strategies are the same in bothmodels in absence of TTPM.

2 Main results

The first focus of this paper is to extend the results obtained in Alfonsi et al. [2,3] and obtain the optimalexecution strategies for LOB with a time varying depth . Doing so, our goal is also to better understandhow this time varying depth may create manipulation strategies. In fact, it was shown in[2] and[3] for1that under some general assumptions on the shape function f, there is an optimal liquidation strategy whichis made only with sell (resp. buy) orders when >0 (resp.


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Before showing the results, it is worth to make further derivations on the expected cost. Let us start withdiscrete strategies. By using the martingale property on S0 and the assumptions on made in Section1.2,we can show easily like in[3] that

C(, T) =S00 + E N

i=0 Di+

Di

(i)xf(x)dx

.

Then, it is easy to check thatN

i=0

Di+Di

(i)xf(x)dx is a deterministic function of(, T)in both volumeimpact reversion and price impact reversion models. We respectively denote byCV(, T)and CP(, T)thisfunction and get:

C(, T) =S00 + E

CM(, T) , (13)where M {V, P} indicates the model chosen. Thus, if the function (x, t) CM(x, t) has a uniqueminimizer on{(x, t) RN RN+1, Ni=1xi = , 0 = t0 tN = T}, the optimal strategy isdeterministic and given by this minimizer. Whenis constant, it is shown in[3]that under some assumptionson f depending on the model chosen, the optimal time grid t is homogeneous with respect to , i.e.ti+1ti

sds = 1N

T0

sds. Instead, there is no such a simple characterization for general , even in the

block-shaped case. Thus, we will focus on optimizing the trading strategy on a fixed time grid t:

t= (t0, . . . , tN), such that0 =t0


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Theorem 2.1. The quadratic form (16) is positive definite if and only if

aiai< 1, i {1, . . . , N } . (17)In this case, the optimal execution problem to liquidate

shares on the time-grid (14) admits a uniqueoptimal strategy which is deterministic and explicitly given by:

0 = KV (t0) 1a11a1a1i = KV (ti)

ai+1

1ai+1ai+1(ai+1 1) + 1ai1aiai

, 1iN 1

N = KV (tN) 1aN1aNaN ,(18)

where

KV = (t0) (1 2a1) + (t1)

1 a1a1 +Ni=2

(ti)(1 ai)21 aiai .

Its cost is given byCV(, t) = 2/(2KV).

This theorem provides an explicit optimal strategy for the large trader. It also gives explicit conditionsthat exclude or create PMS. First, let us assume that

t0, 2t+t0. (19)Then, for any time grid (14),aiai1 and the quadratic form (16) is positive semidefinite since it is a limit ofpositive definite quadratic forms. Thus, the model is PMS free. Conversely let us assume that 2t1+ t1 t1, wherethe large trader buys x >0 at time t1 and sells x at timet2. The cost of such a strategy is given by

CV((0, x, x), t) = x2

2(t2)

et2t1 udu + 1 2e

t2t1 udu

=t2t1

x2

2(t1)((2t1+ t1)(t2 t1) +o(t2 t1))

(20)and is negative whent2 is close enough to t1.

Corollary 2.1. In a block-shaped LOB, modelVdoes not admit price manipulation in the sense of Hubermanand Stanzl if and only if (19) holds.

Let us now discuss this result from the point of view of market makers. A market maker that puts asignificant orders may have an influence on t and t and can increase (resp. decrease) them by respectivelyadding (resp. canceling) an order at a better price or at an existing limit order price. What comes outfrom (19) is that no PMS may arise if one adds limit orders, whatever the way of adding new orders.Instead, PMS can occurs when canceling orders. A different conclusion will hold in the price reversionmodel.

An analogous result to Corollary2.1is stated in a recent paper by Fruth, Schneborn and Urusov [ 9]thathas been published while we were preparing this work. To be precise, results in[9]are given for model Pwith a block-shaped LOB, and the optimal execution strategy is obtained in a continuous time setting. Aswe will see in the next paragraph, models V and Pare mathematically equivalent when the LOB shapeis constant, even though they are different from a financial point of view. By taking a regular time-gridti =

iTN, i = 0 . . . , N , and letting N

+

, we get back the optimal strategy in continuous time (that we

still denote by , by a slight abuse of notations):

0 N+

0 := (T)+

T0

2s(s)

s+2sds

(0)020+0

iNT/N N+

t :=

(T)+T0

2s(s)

s+2sds

(t)

t2t+t

+t

t+t2t+t

, for iN such that

tiNN t0 t

N N+

T := (T)+

T

0

2s(s)

s+2sds

(T)(T+T)T+2T

.

(21)

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The strategy dXt = 00(dt) +

t dt+

TT(dt) with initial trade

0 , continuous trading

t on [t, t+ dt]

for t(0, T) and last trade Tis indeed shown to be optimal in Fruth, Schneborn and Urusov [9] amongthe continuous time strategies with bounded variation. We will show here again this result for more generalLOB shape. The optimal strategy has the following cost:

2

2

(T) +T0

2s(s)2s+s ds

.Besides, this provides a necessary and sufficient condition to exclude transaction-triggered price manipulation.

Corollary 2.2. In a block-shaped LOB, modelVdoes not admit transaction-triggered price manipulation ifand only if

t0, t+t0, and

t2t+t

+t

t+t2t+t

0. (22)

The first condition comes from the last trade and implies (19) since t 0. It can be interpretedsimilarly as condition (19) from market makers point of view. The second condition in (22) comes fromthe intermediate trades and brings on the derivatives of and . It is harder to get an intuitive idea ofits meaning from a market makers point of view. Last, let us mention that we can show that the optimal

strategy on the discrete time-grid (14) is made with nonnegative trades if one has(17) and

1 ai1 aiai ai+1

1 ai+11 ai+1ai+1 , i {1, . . . , N 1} andaN 1. (23)

Condition (22) can be seen as the continuous time limit of condition (23).Let us give now an illustration of the optimal strategy with a time-varying depth. We consider the case

of a time-varying depth(t) =0+ cos(2t), with 0> 1,

which corresponds to a one-day periodic function with high values at the beginning and at the end of the day.We can show thatt 2

201and with a constant resilience, there is no PMS as soon as2 2

2010.

Figure1shows the optimal execution strategy (18) with a value0that exclude PMS but allows TTPM. The

optimal strategy to buy 50 shares consists in buying almost 95 shares and selling 45 shares, which roughlytreble the traded volume.

2.1.2 Price impact reversion model

Whenf1, the deterministic cost function Ni=0 Dti+Dti (ti)xf(x)dx is given byCP(, t) =

Nn=0

(tn)

Dtn+Dtn

xf(x)dx=Ni=0

i2

i

(ti)+ 2

j


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

15

10

5

0

5

10

15

20

Optimal Execution Strategy

Time

Ordersize

Figure 1: Optimal execution strategy to buy 50 shares on a regular time grid, with N = 20, = 1,

(t) = 4 + cos(2t)(plotted in dashed line). In solid line is plotted the function t

t2t+t

+ t

t+t2t+t

.

and the optimal execution problem in Model P with resilience , LOB depth (t) and time-grid t is the

same as the optimal execution problem in Model V with resilience , LOB depth (t) and time-grid t. Weimmediately get the following results.

Theorem 2.2. The quadratic form (24) is positive definite if and only if

aiai< 1, i {1, . . . , N } (26)

In this case, the optimal execution problem to liquidate

shares on the time-grid (14) admits a uniqueoptimal strategy which is deterministic and explicitly given by:

0 = KP (t0) 1a11a1a1 .i = KP (ti)

ai1aiai

(ai 1) + 1ai+11ai+1ai+1

, 1iN 1N = KP (tN) 1aN1aNaN

(27)

where

KP =

(tN)(1

2aN) +(tN1)

1 aNaN +

N2i=0 (ti)

(1

ai+1)

2

1 ai+1ai+1 .Its cost is given byCP(, t) = 2/(2KP).

By taking a regular time-grid ti= iTN, i= 0 . . . , N , and lettingN+, we get the optimal strategy in

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continuous time:

0 N

0 := (0)+

T

0

2s(s)

2ssds

(0) 00200

iNT/N

N

t := (0)+

T

0

2s(s)

2ssds

(t)

tt2tt

+t

tt2tt

, foriN such that

T iNN

t0

t

N N

T :=

(0)+T0

2

s

(s)

2ssds

(T) T

2TT.

(28)

The strategy with initial trade 0 , continuous tradingt on[t, t + dt]for t(0, T)and last tradeT is shown

to be optimal in Fruth, Schneborn and Urusov [9] among the continuous time strategies with boundedvariation, and has the following cost:

2

2

(0) +T0

2s(s)2ss

ds .

Corollary 2.3. In a block-shaped LOB, modelPdoes not admit price manipulation in the sense of Hubermanand Stanzl if and only if

t0, 2t t0. (29)It does not admit transaction-triggered price manipulation if and only if

t0, t t0, and

t t2t t

+t

t t2t t

0. (30)

The first condition in (30) comes from the initial trade while the second comes from intermediate trades.From market makers point of view, (29) and the first condition in (30) give different conclusions frommodel V. A significant market maker will not create manipulation strategy if he puts orders at a betterprice (which increases ) or cancels orders at existing prices (which decreases ). Instead, he may createmanipulation strategies if he piles orders at existing prices, or if he cancels orders that are among the bestoffers. The second condition of (30) brings on the dynamics of and and it is more difficult to give itsheuristic meaning in terms of trading. Last, let us mention that the optimal strategy in discrete time givenby Theorem2.2 is made only with trades of same sign if, and only if, one has ( 26) and

1 ai+11 ai+1ai+1 ai

1 ai1 aiai , i {1, . . . , N 1} and a1 < 1. (31)

2.2 Results for general LOB shape

We extend in this section the results obtained on the optimal execution for block-shaped LOB to moregeneral shapes. In particular, the necessary and sufficient conditions that we have obtained to excludeTTPM (namely(22)for model V and (30) for model P) are still sufficient conditions to exclude TTPM fora wider class of shape functions. From a mathematical point of view, the approach is the same. We firstcharacterize the optimal strategy on a discrete time grid, by using Lagrange multipliers. Then, one can guessthe optimal continuous time strategy, and we prove its optimality by a verification argument.

2.2.1 Volume impact reversion model

We first introduce the following assumption that will be useful to study the optimal discrete strategy.

Assumption 2.1. 1. The shape functionf satisfies the following condition:

fis nondecreasing onR and nonincreasing onR+

2.t0, t+t0.

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We remark that when the LOB shape does not evolve in time (t = 0), the second condition is satisfiedand we get back the assumption made in Alfonsi, Fruth and Schied [2]. We define

xR, hV,i(x) = F1(x) aiF1 (aix)

1 ai , 1iN. (32)

Theorem 2.3. Under Assumption 2.1, the cost function CV(, t) is nonnegative, and there is a unique

optimal execution strategy that minimizesCV over{RN+1, Ni=0i = }. This strategy is given asfollows. The following equation

Ni=1

(ti1)(1 ai)h1V,i() +(tN)F() =

has a unique solution R, and0 = (t0)h

1V,1() ,

i = (ti)(h1V,i+1() aih1V,i()), 1iN 1,

N = (tN)F() (tN1)aNh1V,N() .The first and the last trade have the same sign as . Besides, if the following condition holds

1ai

1 ai1 ai

1 ai+11 ai+1 , (33)

the intermediate tradesi, 1iN 1, have also the same sign as .This theorem extends the results of[2], where is assumed to be constant. In that case, (33) is satisfied

and all the trades have the same sign. Condition (33) is interesting since it does not depend on the shapefunction, but it is more restrictive than the condition (23) for the block-shape case (see Lemma 3.4 for(33) = (23)). In fact, the continuous time formulation is more convenient to analyze the sign of thetrades. Under Assumption 2.1, we will show that no transaction-triggered price manipulation can occurwith the same condition (22) as for the block-shape case.

When stating the optimal continuous-time strategy, we slightly relax Assumption 2.1. This is basicallydue to the argument of the proof that relies on a verification argument. Instead, our proof in the discretecase relies on Lagrange multipliers which requires to show first that the cost function has a minimum, and

we uset+t0 for that. We introduce the following functionhV,t(x) =F

1(x) +t+t

t

x

f(F1(x)). (34)

We will show that no PMS exists and that there is a unique optimal strategy if these functions fort[0, T]are bijective on R with a positive derivative. If Assumption2.1holds, this condition is automatically satisfied.

Theorem 2.4. Letf C1(R). We assume that for t [0, T], hV,t is bijective on R, such that hV,t > 0.Then, the cost function CV(X) is nonnegative, and there is a unique optimal admissible strategyX thatminimizesCV. This strategy is given as follows. The equation T

0

(t)th1V,t()dt+(T)F() = (35)

has a unique solution R and we sett = h1V,t(). The strategydX

t =

00(dt) +

tdt+

TT(dt) with

0 = (0)0,

t = (t)

dtdt

+ (t+t)t

,

T = (T)(F() T),is optimal. The initial trade0 has the same sign as .

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Thus, a sufficient condition to exclude price manipulation strategies is to assume that hV,t is bijectivewithhV,t >0. We have a partial reciprocal result: there are PMS as soon as h

V,t1

(0)< 0 for some t1 0.Indeed, in this case we consider the following round trip on the time grid t= (0, t1, t2) with t2 > t1, wherethe large trader buys x >0 at time t1 and sells x at time t2. The cost of such a strategy is given by

CV((0, x,

x), t) = (t1)G x

(t1) +(t2) Gx(e

t2t1 sds 1)

(t2) G

xet2t1 sds

(t2)

= (t1)

t1G

x

(t1)

+ (t1+ t1)

x

(t1)F1

x

(t1)

(t2 t1) +o(t2 t1).

The derivative ofx t1G(x) + (t1+ t1)xF1(x)is t1hV,t1(x), which has the opposite sign ofx near0since hV,t1(0) = 0 and hV,t1(0) < 0 by assumption. Thus, we have C

V((0, x, x), t) < 0 for x and t2 t1small enough.

Now, let us focus on the sign of the trades given by the optimal strategy. Without further hypothesis,the condition t0 typically involves the shape function f. However, under Assumption2.1,we can showthat transaction-triggered strategy are excluded under the same assumption as for the block-shape case.

Corollary 2.4. Let f C1. Under Assumption2.1, the function hV,t isC1(R), bijective on R, and suchthath V,t > 0. Thus, the result of Theorem2.4holds and the last trade

Thas the same sign as

.Besides, if (22) also holds, t has the same sign as for any0< t < T, which excludes TTPM.Let us now focus on the example of a power-law shape: we assume that

f(x) =|x|, >1.

In this case,F(x) = sgn(x) |x|+1

+1 is well-defined and satisfies(4). We haveF1(x) = sgn(x)(+ 1)

1+1 |x| 1+1

andhV,t(x) = sgn(x)(+ 1) 1+1 |x| 1+1

t(2+)+tt(1+)

. Thus, hV,t is bijective and increasing if, and only if:

t(2 +) +t > 0.

In this case, we have

h1

V,t(x) =

1

+ 1 Kt()sgn(x)|x|+1

with Kt() = t(1 +)

t(2 +) +t1+

.

In this case, we have by Theorem 2.4 that

0 =

T

0 (t)tKt()dt+(T)

(0)K0(),

t =

T

0 (t)tKt()dt+(T)

(t)dKt()dt + (t+t)Kt()

T =

T0 (t)tKt()dt+(T)

(T)(1 KT())(36)

is the unique optimal strategy. For = 0, we get back (21). If we only assume thatt(2 + ) + t 0,we still haveCV(X)0 for any admissible strategyX. The cost CV(X) is indeed continuous with respectto the resilience, and is the limit of the cost associated to resilience t + , 0. On the contrary, ift(2 +) +t< 0, we have hV,t(0)< 0 and there is a PMS as explained above.

Corollary 2.5. Whenf(x) =|x|, modelVdoes not admit PMS if, and only ift0, t(2 +) +t0.

It does not admit transaction-triggered price manipulation if and only if

t0, t+t0, and

t(1 +)

t(2 +) +t

+t

t+t

t(2 +) +t

0.

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These conditions comes respectively from the nonnegativity of the last and intermediate trades. For givenfunctionst and t, the no PMS condition will be satisfied for t[0, T] when is large enough. This canbe explained heuristically. When increases, limit orders become rare close to S0t and dense away from S

0t,

which creates some bid-ask spread. One has then to pay to get liquidity, and round trips have a positivecost. Instead, when is close to1 it is rather cheap to consume limit orders, which may facilitate PMS.In Figure2, we have plotted the optimal strategy for =0.3 and = 1 with the same parameters as inFigure1 for the Block shape case. We can check that the no PMS condition is satisfied in both cases.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

50

40

30

20

10

0

10

20

30

40

50


Time

Ordersize

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

15

10

5

0

5

10

15

20


Time

Ordersize

Figure 2: Optimal execution strategy to buy 50 shares on a regular time grid, with N = 20, = 1,(t) = 4 + cos(2t)(plotted in dashed line) and =0.3 (left) or = 1 (right). In solid line is plotted thefunctiont

t(1+)t(2+)+t

+t

t+tt(2+)+t

(this function is well-defined but out of the graph for=0.3).

2.2.2 Price impact reversion model

The results that we present for model P are similar to the one obtained for model V. We first solve the

optimal execution problem in discrete time. From its explicit solution, we then calculate its continuous timelimit and check by a verification argument that it is indeed optimal. Doing so, we get sufficient conditionsto exclude PMS and TTPM. In particular, condition (30) that excludes PMS and TTPM for block-shapeLOB also excludes PMS and TTPM for a general LOB shape satisfying Assumption 2.2 below.

To study the optimal discrete strategy, we will work under the following assumption.

Assumption 2.2. 1. The shape functionf isC1 and satisfies the following condition:

f is nonincreasing onR and nondecreasing onR+

2.t0, t t > 0.

3. x

x f

(x)f(x) is nondecreasing onR, nonincreasing onR+.

The monotonicity assumption made here is the opposite to the one made in Assumption2.1for modelV.This choice is different from the one made in Alfonsi et al. [2, 3]. It is in fact more tractable from amathematical point of view, especially here with a time-varying LOB.

Theorem 2.5. Under Assumption 2.2, the cost function CP(, t) is nonnegative, and there is a unique

optimal execution strategy that minimizesCP over{RN+1, Ni=0i = }. This strategy is given as

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follows. The following equation

Ni=1

(ti1)

F

h1P,i()

ai

(ti)

(ti1)F(h1P,i())

+(tN)F() =

has a unique solution R, and

0 = (t0)F

h1P,1()

a1

,

i = (ti)

F

h1P,i+1()

ai+1

F(h1P,i())

, 1iN 1,

N = (tN)[F() F(h1P,N())].The first and the last trade have the same sign as .

We now state the corresponding result in continuous time and set:

xR, hP,t(x) = x

1 +

t

t 1 + xf(x)

f(x) t

. (37)

Theorem 2.6. Letf C2(R). We assume that one of the two following conditions holds.(i) Fort[0, T], t

1 + xf

(x)f(x)

t > 0 for anyxR andhP,t is bijective onR, such thathP,t(x)> 0,

dx-a.e.

(ii) Fort[0, T], t

1 + xf(x)

f(x)

t < 0 andt

2 + xf

(x)f(x)

t > 0 for anyxR, andhP,t is bijective

onR, such thathP,t(x)< 0, dx-a.e.

Then, the cost function CP(X) is nonnegative, and there is a unique optimal admissible strategyX thatminimizesCP. This strategy is given as follows. The equation T

0

(t)[th1P,t()f(h

1P,t()) tF(h1P,t())]dt+(T)F() = (38)

has a unique solution R and we sett = h1P,t(). The strategydXt =00(dt) +t dt+TT(dt) with0 = (0)F(0),

t = (t)f(t)

dtdt

+tt

,

T = (T)(F() F(T)),is optimal. The initial trade0 has the same sign as .

In particular, there is no PMS in model Pas soon as Assumptions (i) or (ii) hold. Conversely, let us

assume that t1

2 + xf

(x)f(x)

t1 t1, and consider that the large trader buysx >0 at timet1 and sellsx at timet2.The cost of such a round trip is

CP((0, x, x), t)= (t1)G

x

(t1)

+(t2)

G

F

e

t2t1 sdsF1

x

(t2)

x

(t2)

F

e

t2t1 sdsF1

x

(t2)

= (t1)

t1 F

F1

x

(t1)

+t1F

1

x

(t1)

2f

F1

x

(t1)

(t2 t1) +o(t2 t1).

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The derivative ofx t1 F(x) + t1x2f(x)is xf(x)

t1

2 + xf

(x)f(x)

t1

and has the opposite sign ofx

near0. Thus, CP((0, x, x), t) is negative when t2 is close to t1 andx is small enough, which gives a PMS.Corollary 2.6. Letf C2(R). Under Assumption2.2, the functionhP,t isC1(R), bijective onR and suchthath P,t > 0. Thus, the result of Theorem2.6holds and the last trade

Thas the same sign as .

Besides, if (30) also holds, t has the same sign as for any0< t < T, which rules out TTPM.As for model V, we consider now the case of a power-law shape f(x) =|x|. We can apply the results of

Theorem2.6 in this case. We can also notice from (9) thatdEt = (t t(1 +))Etdt. Therefore, model Pwith resilience t is the same as model V with resilience t = t(1 +) t.Corollary 2.7. Whenf(x) =|x|, modelPdoes not admit PMS if, and only if

t0, t(2 +) t0.It does not admit transaction-triggered price manipulation if and only if

t0, t(1 +) t0, and

t(1 +) tt(2 +) t

+t

t(1 +) tt(2 +) +t

0.

3 Proofs

3.1 The block shape case

Proof of Theorem2.1: The quadratic form (16) is given byCV(, t) = 12TMV, withMVi,j =

exptjti sds

(titj) ,

0i, jN. Let us assume that aiai < 1, i {1, . . . , N }. Then, we can define the following vectors:

y0 = e0

(t0), yi = aiyi1+

ei(ti)

1 aiai, 1iN

where e0. . . eN denote the canonical basis of RN+1. We haveMVij = yTi yj . We introduce Y the upper

triangular matrix with columns y0, . . . ,yN. By assumption, it is invertible and so is M=YTY. Conversely,

ifMV is positive definite, the minors

det((MVi,j)0i,jn) = 1

(t0)

ni=1

1

(ti)(1 aiai), 1nN

are positive, which gives (17).Let us turn to the optimization problem. One has to minimize CV(, t) under the linear constraintNi=0i = , which gives

= 1T (MV)

11

MV

11, (39)

where 1 RN+1 is a vector of ones. Since Y is upper triangular, it can be easily inverted and we cancalculate explicitly

MV

11 and get (18).

3.2 General LOB shape with model V

Let us introduce some notations. For the time grid t given by(14), we introduce the next quantities:

k :=

tktk1

sds, k= 1, . . . , N . (40)

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We can write the cost function (13) as follows

CV(, t) =Nn=0

(tn)

G

En+n

(tn)

G

En(tn)

, (41)

where we use the following notations (observe that En= an(En1+n1))

E0 = 0, En=n1i=0

ie

nk=i+1k , 1nN.

Lemma 3.1. We have CV

N=F1

EN+N(tN)

and, for i= 0, . . . , N 1,

CV

i ai+1 C

V

i+1=F1

Ei+i

(ti)

ai+1F1

Ei+1(ti+1)

. (42)

Proof. Let us first observe that Eni = 0, ifin, and Eni =e

nk=i+1k ifi < n. Thus, we get by using

thatG =F1:

CV

i= F1

Ei+i

(ti)

+

Nn=i+1

en

k=i+1k

F1

En+n

(tn)

F1

En(tn)

= F1

Ei+i(ti)

ei+1F1

Ei+1(ti+1)

+ ei+1

F1

Ei+1+i+1

(ti+1)

+

Nn=i+2

e

nk=i+2k

F1

En+n

(tn)

F1

En(tn)

= F1

Ei+i(ti)

ai+1F1

Ei+1(ti+1)

+ai+1

CV

i+1.

Lemma 3.2. Under Assumption2.1, we obtain the next conclusions.

1. For i {1, . . . , N }, the function hV,i defined in (32) is an increasing bijection on R that satisfiessgn(x)hV,i(x)

1aiai1ai

F1(x).

2. If (33) holds, then we havesgn(x)h1V,i+1(x)sgn(x)aih1V,i (x) for i {1, . . . , N 1}.3. sgn(x)F(x)sgn(x)aNh1V,N(x).

Proof. 1. Since the resilience t is positive, we have 0 < ai < 1, and ai 1 since t +t 0 byAssumption2.1. We then get

hV,i(x)

x =

1

1 ai

1

f(F1(x)) aiai

f(F1(aix))

1 aiai

1 ai1

f(F1(x))>0

because f is nondecreasing on R and nonincreasing on R+, and F1 is increasing.

2. We set f(x) = (F1)(x) = 1/f(F1(x)): this function is positive, nonincreasing on R and nonde-creasing on R+. Let0 and y = h1V,i+1(). We note thaty0 becausehV,i+1(0) = 0and hV,i+1 isincreasing by the first point of this lemma. Thus, we have that

= F1(y) ai+1F1(ai+1y)

1 ai+1= F1(ai+1y) +

F1(y) F1(ai+1y)1 ai+1

= F1(ai+1y) + 1

1 ai+1

yai+1y

f()dF1(y) +1 ai+11 ai+1 y

f(y) =: gi+1(y)

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Hence, we obtain that gi+1 is increasing on Rand then, yg1i+1(). Letz = aih1V,i()0. We have:

=F1

zai

aiF1(z)

1 ai

= F1

(z) +

F1 zai F1(z)1 ai

= F1(z) + 1

1 ai

zai

z

f()dF1(z) +

1ai

1

1 ai zf(z) =: gi(z)

Therefore, if (33) holds, we get that gi+1(x) gi(x) for all x 0. Then, we have g1i+1(x) g1i (x),and therefore

yg1i+1()g1i ()z.The same arguments for 0 giveyg1i+1()g1i ()z .

3. Using the above definition, we havesgn(x)gN(x)sgn(x)F1(x), and therefore we get

sgn()F()

sgn()g1

N

()

sgn()z = sgn()aNh1

V,N

() .

Lemma 3.3. Let a (0, 1) and b > 0 such that ab 1. We have G(x) 1bG(abx) 0 for x R, andG(x) 1bG(abx) |x|+ +.

Proof. Since G is convex (G =F1 is increasing) andG(0) = 0, G(abx)abG(x). Ifb >1, we then haveG(x) 1bG(abx)G(x)(1 a) which gives the result. Ifb1, we have

G(x) 1b

G(abx) =

x0

F1(u)du 1b

abx0

F1(u)du=

x0

F1(u)du ax0

F1(bv)dv

= x

ax

F1(u)du+ ax

0F1(u) F

1(bu) du |x|(1 a)F1(

|ax

|)

|x|+ .

Proof of Theorem 2.3: We rewrite the cost function (41) to minimize as follows:

CV(, t) =Nn=0

(tn)

G

En+n

(tn)

G

En(tn)

= (tN)G

Ni=0ie

Nk=i+1k

(tN)

(0)G(0)

+

N1n=0

(tn)G

ni=0ie

n

k=i+1k

(tn)

(tn+1)G

en+1

ni=0ie

n

k=i+1k

(tn+1)

We define the linear map T : RN+1 RN+1 by(T )n= n

i=0ie

nk=i+1k

(tn) , so that

CV(, t) = (tN)G((T )N) +

N1n=0

[(tn)G((T )n) (tn+1)G (an+1(T )n)] . (43)

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Let us observe thatTis a linear bijection. By Lemma3.3we get thatCV(, t)0andCV(, t) ||+

+,which gives the existence of a minimizer over , s.t.

Ni=0i = . Thus, by using(42), there must be a

Lagrange multiplier such that

= hV,i+1Ei+

i

(ti) , i= 0 . . . N 1, and= F1

EN+

N

(tN) . (44)We have

Ei+i

(ti) =h1V,i+1()and then Ei+1 = (ti)ai+1h

1V,i+1(), for0iN 1. Thus, we get

0 = (t0)h1V,1() ,

i = (ti)h1V,i+1() (ti1)aih1V,i() , 1iN 1,

N = F()(tN) (tN1)aNh1V,N()Furthermore, we note that

Ni=0

i = =(t0)(1 a1)h1V,1() +. . .+(tN1)(1 aN)h1V,N() +F()(tN).

By Lemma3.2The right side is an increasing bijection on R, and we deduce that there is only one Rwhich satisfies the above equation. This give the uniqueness of the minimizer . Moreover, the functionsF1 andhV,i vanish in 0, andhas the same sign as , which gives that0 andNhave the same sign as

by Lemma3.2. Besides, if (33) holds, the tradesi have also the same sign as .

Let us now prepare the proof of Theorem 2.4 and assume that hV,t is bijective increasing. We introducefor0tT,

CV(t , T , E t, Xt) = (t)

G(t) G

Et(t)

+

Tt

F1(u)udu+(T)[G(F()) G(T)], (45)

where

R, s.t. Et+ T

t

(u)uh1V,u()du+(T)F() =Xt, (46)

u= h1V,u(), u= (u)[

dudu

+ (u+u)u]. (47)

Let us observe that Tt (u)uh1V,u()du+(T)F() is increasing an bijective on R, and (46) admits aunique solution. The function CV(t , T , E t, Xt)denotes the minimal cost to liquidate Xt shares on the timeinterval[t, T] given the current state Et. In particular, we observe that

CV(T , T , E T, XT) = (T)

G

ET XT

(T)

G

ET(T)

,

which is the cost of selling XTshares at time T. Besides, an integration by parts gives that

CV(t , T , E t, Xt) =(t)G

Et(t)

+

Tt

(u)

(u+u)F1(u)u uG(u)

du+(T)G(F()). (48)

The function(u + u)F1()uG()is nonnegative since it vanishes at 0, and its derivative is equaltouhV,u()that has the same sign as . Since G0, we get:

CV(0, T, 0, ) 0. (49)

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Formula (45) can be guessed by simple but tedious calculations: one has to consider the associateddiscrete problem on a regular time-grid and then let the time-step going to zero. We do not present thesecalculations here since we will prove directly by a verification argument that this is indeed the minimal cost.

Proof of Theorem 2.4: Let (Xt, 0 t T+) denote an admissible strategy that liquidates . Weconsider(Et, 0tT+)the solution ofdEt = dXt tEtdt, t the solution of (46) andt = h1V,t(t). Weset

Ct = t0

F1

Es(s)

dXcs+

0s 0. Then, we have

Ct = (t)

G

Et+ Xt

(t)

G

Et(t)

+CV(t+, T , E t+, Xt+) CV(t , T , E t, Xt).

SinceEt = Xt, the solutiont of (46) is also the solution ofEt++Tt

(u)uh1V,u(t)du + (T)F(t) =

Xt+, and then Ct = 0. Now, let us calculate dCt. We set

C(t , T , E t, Xt, v) = (T)G(F(v)) (t)G Et(t)+

Tt

(u)

(u+u)F1(h1V,u(v))h

1V,u(v) uG(h1V,u(v))

du.

Then, we have from (48):

dCt = F1

Et(t)

dXct (t)G

Et(t)

dt F1

Et(t)

(dXct (t+t)Etdt)

(t)(t+t)F1(t)tdt+(t)G(t)dt+ C

v(t , T , E t, Xt, t)dt.

Since (T)f(t) + Tt (u)u(h1V,u)(t)du dt (t)th1V,t(t)dt= d(Et

Xt) =

tEtdtand

vC(t , T , E t, Xt, v) = (T)vf(v) +

Tt

(u)u(h1V,u)

(v)

F1(h1V,u(v)) +

u+uu

h1V,u(v)

f(h1V,u(v))

du

= v

(T)f(v) +

Tt

(u)u(h1V,u)

(v)du

,

we finally get

dCt = (t)

(t+t)

Et(t)

F1

Et(t)

tF1(t)

+t

G(t) G

Et(t)

+thV,t(t)

t Et

(t)

dt

:= (t)t(t)dt. (50)

We havet() =(t+ t)

F1() + f(F1())

+ tF1()+ thV,t()+ thV,t()( Et(t)) = thV,t()(

Et(t)). Sinceh

V,t > 0, t vanishes at =

Et(t) , and is positive for= Et(t) .

Thus, ifXis an optimal strategy, we necessarily have t = Et(t) , dt-a.e. Then, we get by differentiat-

ing

Xt Et+Tt (u)uh

1V,u(t)du+(T)F(t)

= 0 that

Tt (u)u(h

1V,u)

(t)du+(T)f(t)

dt = 0,

which givesdt = 0since(h1V,u)

>0 and f >0. Thus, we get that t= whereis the solution of (35). In

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particular, we get X0 =E0+ =(0)h1V,0(0) = X

0 and then X=X

, which gives the uniqueness of theoptimal strategy. Last we observe that has the same sign as and thus0 has the same sign as .

Proof of Corollary 2.4: Since t+t0 and xf(F1(x))0 by Assumption2.1,we have

hV,t(x) =t+ 2

tt

1

f(F1(x))t+

tt

xf(F1(x))

f(F1(x))3 >0.

Also, we havesgn(x)hV,t(x)sgn(x)F1(x)and thensgn(x)h1V,t(x)sgn(x)F(x), which gives that the lasttradeThas the same sign as . Then, we have dtdt = 1h

V,t(t)

dhV,tdt (t)and thus

t = (t)t

hV,t(t)

d (t/t)

dt

1

f(F1(t))+ (t+t)h

V,t(t)

= (t)t

hV,t(t)

1

tf(F1(t))

tt tt

t+ (t+t)(2t+t)

(t+t)

2

t

tf(t)

f(F1(t))3

is nonnegative if (22)holds sincehV,t > 0 and tf(t)0.

Lemma 3.4. We have(33) = (23) ift+t0, t0.Proof. We have

(33) 1ai

1 ai1 ai

1 ai+11 ai+1 (1 ai+1) ai(1 ai+1)ai(1 ai) aiai+1(1 ai)

ai+1(1 ai) + 1ai

(1 ai+1)1 ai+ 1 ai+1.

Sinceai+11, we get1 ai+ 1 ai+1 = 1 aiai+1 + (1 ai)(1ai+1)1 aiai+1 + ai+1(1 ai)(1ai+1).Thus, (33) implies that:

ai+1(1 ai) + 1ai

(1 ai+1)1 aiai+1+ ai+1(1 ai)(1 ai+1)1 ai+aiai+1ai aiai+1ai+1 aiai+1ai+1+aiaiai+1ai+1 ai+1ai+1

(1 ai) (1 ai+1ai+1)ai+1(1 ai+1) (1 aiai)(23).

3.3 General LOB shape with model P

We first focus on discrete strategies on the time grid t such as (14). We introduce the following shorthandnotationDn= Dtn for0nNand have

D0 = 0, Dn= anF1

n1(tn1)

+ F(Dn1)

, 1nN.

We can write the cost function (13) as follows:

CP(, t) =Nn=0

(tn)

Dtn+Dtn

xf(x)dx=Nn=0

(tn)

G

(tn)F(Dn) +n

(tn)

G(F(Dn))

. (51)

We begin with the following lemmas that we use to characterize the critical points of the optimizationproblem.

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Lemma 3.5. Fori = 0, . . . , N 1, we have the following equations:CP

i=F1

i(ti)

+ F(Di)

+ ai+1

f(Di+1)

f

F1 i(ti)

+ F(Di) CP

i+1 Di+1

.

Proof. First, we have Dni = 0 for in, and the following recursive equations:Dn

n1=

an

(tn1)f

F1( n1(tn1) + F(Dn1)) , Dn

i=

ai+1f(Di+1)

f

F1( i(ti) + F(Di)) Dn

i+1for1in 2.

From (51), we get:

CP

i= F1

i(ti)

+ F(Di)

+

Nn=i+1

F1

F(Dn) +

n(tn)

Dn

f(Dn)

Dni

= F1

i(ti)

+ F(Di)

+

ai+1f(Di+1)

f

F1( i(ti) +F(Di)) F1(F(Di+1) + i+1

(ti+1)) Di+1

+ ai+1f(Di+1)

f

F1

( i(ti) + F(Di))

CP

i+1

F1 i+1(t

i+1)

+ F(Di+1) ,which gives the result.

Lemma 3.6. Under Assumption2.2, we have that:

1. The functionxxf(x) is increasing onR (or equivalently, F is convex).

2. We havefxai

aif(x)> 0, i= 1, . . . , N .

3. The function

xR, hP,i(x) =x 1ai

f( xai ) aif(x)

f

xai

aif(x)

is well-defined, bijective increasing and satisfiessgn(x)hP,i(x) |x|.Proof. 1. We have (xf(x)) >0 since xf(x)0 by Assumption2.2.2. We have for xR,

(ti1)f(x

ai) (ti)aif(x)(ti1)f(x)(1 ai)> 0

because fxai

f(x) and ai< 1 by Assumption2.2.

3. The function hP,i is well-defined thanks to the second point. We have sgn(x)hP,i(x) |x| since

hP,i(x) =x

1 + a1i

1 ai f(x)f( xai)

,

and it is sufficient to check that f(x)/f(x/ai)is nondecreasing on R+and nonincreasing on R. We calculate f(x)

fxai

=f(x)f

xai

1ai f(x)f

xai

fxai

2 .This is nonnegative on R+ and nonpositive on R if and only if

xf(x)f(x) xf

(x/ai)aif(x/ai)

forxR, which holds byAssumption2.2 since|x| |x|/ai.

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Proof of Theorem 2.5: We remark that the cost (51) can be written as follows:

CP(, t) = (tN)F

F1

F(DN) +

N(tN)

+N1

n=0 (tn) FF1

F(Dn) + n(tn)

(tn+1)

(tn)Fan+1F

1

F(Dn) + n(tn) .

Since F is convex by Lemma3.6 and F(0) = 0, we have F(an+1x)an+1F(x), forxR and thus

CP(, t) (tN)F

F1

F(DN) + N(tN)

+

N1n=0

(tn)F

F1

F(Dn) +

n(tn)

(1 an+1).

In particular CP(, t) 0, since F 0 and an+1 < 1 by Assumption (2.2). Besides, by setting T() = 0(t0)

, D1+ 1(t1)

, . . . , DN+ N(tN)

, we can easily check that|T()|

||++, which gives immediately

thatCP(, t) ||+

+since F(x) |x|+

+.Thus, there must be at least one minimizer ofCP(, t) on{ RN+1, Ni=0i = }, and we denote

bya Lagrange multiplier such that CP

i=. By Lemma3.5 we obtain:

=hP,i+1(Di+1), i= 0, . . . , N 1.

We also have CP

N=F1

F(DN) +

xN(tN)

= , and we get (i= 1, . . . , N 1):

0 =(t0)F

h1P,1()

a1

, i =(ti)

F

h1P,i+1()

ai+1

F

h1P,i()

, N =(tN)

F() F(h1P,N())

.

Besides, we have

(tN)F() +Ni=1

(ti1)

F

h1P,i()

ai

(ti)

(ti1)F(h1P,i())

= . (52)

SinceF is increasing bijective on R and the function yF(y) (ti)(ti1)F(aiy) is increasing (its derivative

is positive by Lemma 3.6), there is a unique solution to (52), and has the same sign as . Thus is the unique optimal strategy. Moreover, the initial and the last trade have the same sign as sincesgn()hP,N() ||.

We now prepare the proof of Theorem2.6. For sake of clearness, we will work under assumption (i)and

assume that t

1 + xf

(x)f(x)

t > 0 for any x R and that hP,t is bijective and increasing. However, a

close look at the proof below is sufficient that the same arguments also work under assumption (ii).Contrary to model V, it is more convenient to work with the process D rather than E(both are related

byDt = F1(Et/(t)). We introduce for 0tT,

CP

(t , T , Dt, Xt) = (t)

G(t) F(Dt) + T

t uudu+(T)[F() G(T)], (53)

where

R,s.t. Et+ Tt

(u)

uh1P,u()f(h

1P,u()) uF

h1P,u()

du+(T)F() =Xt, (54)

u= h1P,u(), u = (u)f(u)[

dudu

+uu]. (55)

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Let us observe thatxuxf(x)uF(x)is increasing: its derivative is equal to f(x)

u

1 + xf(x)

f(x)

u

and is positive by assumption. Therefore, the left hand side of (54) is an increasing bijection on R andthere is a unique solution to (54). The function CP(t , T , Dt, Xt) represents the minimal cost to liq-uidate Xt shares on [t, T] given the current state Dt. We have in particular that C

P(T , T , DT, XT) =

(T)

G

ETXT(T)

G ET(T)

, which is the cost of selling XT shares at time T. Besides, an integration

by parts gives that

CP(t , T , Dt, Xt) =(t)F(Dt) + Tt

(u)

uf(u)2u uF(u)

du+(T)F(). (56)

The function uf()2 uF() is nonnegative: it vanishes for = 0 and its derivative is equal tof()

u

2 + f

()f()

u

and has the same sign as by assumption. Since F 0, this gives

CP(0, T, 0, ) 0. (57)Proof of Theorem 2.6: Let (Xt, 0 t T+) denote an admissible strategy that liquidates . We

consider (Et, 0 t T+) the solution of dEt = dXt + tEtdtt(t)f(F1(Et/(t)))F1(Et/(t))dt,Dt = F1(Et/(t)), t the solution of (54) andt = h

1P,t(t). We set

Ct = t0

DsdXcs +

0s 0. Then, we have

Ct = (t)

G

Et+ Xt

(t)

G

Et(t)

+CP(t+, T , Dt+, Xt+) CP(t , T , Dt, Xt).

SinceEt = Xt, we have t = t+ from (54)and then Ct = 0 since F(Dt) = G(Et/(t)). Now, let uscalculatedCt. We set

C(t , T , Dt, Xt, v) = (T)F(v)

(t)F(Dt) + T

t

(u) uf(h1P,u

(v))h1

P,u(v)2

uF(h

1

P,u(v)) du.

SincedDct =tDtdt+ dXct

(t)f(Dt), we have from (56):

dCt = DtdXct (t)F(Dt)dt+(t)tf(Dt)D2t dt DtdXct (t)[tf(t)2t tF(t)]dt

+C

v(t , T , Dt, Xt, t)dt.

Sinced(Et Xt) = (t) [tF(Dt) tDtf(Dt)] dt, we get from (54) Tt

(u)(h1P,u)(t)

(u u)f(h1P,u(t)) +uh1P,u(t)f(h1P,u(t))

du+(T)f(t)

dt (58)

(t) th1P,t(t)f(h1P,t(t)) tFh

1P,t(t) dt= (t) [tF(Dt) tDtf(Dt)] dt.

On the other hand, we have

vC(t , T , E t, Dt, v) = (T)vf(v) +

Tt

(u)(h1P,u)(v)h1P,u(v)

(2u u)f(h1P,u(v)) +uh1P,u(v)f(h1P,u(v))

du

= v

(T)f(v) +

Tt

(u)(h1P,u)(v)

(u u)f(h1P,u(v)) +uh1P,u(v)f(h1P,u(v))

du

,

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and we get Cv(t , T , Dt, Xt, t)dt = (t)t[t(F(Dt) F(t)) +t(tf(t) Dtf(Dt))]. We finally obtain:

dCt = (t)t(t)dt, with (59)

t() = t(F() F(Dt)) +t(D2t f(Dt) 2f()) +hP,t() (t(F(Dt) F()) +t(f() Dtf(Dt))) .We have t(Dt) = 0 and get that

t() = h

P,t() [t(F(Dt) F()) +t(f() Dtf(Dt))] by simple

calculations. On the one hand, we havehP,t

()> 0. On the other hand, the bracket is positive on > Dtand negative on < Dtsince its derivative is equal to (tt)f()+ tf(), which is positive by assumption.

Thus,Dt is the unique minimum oft: t(Dt) = 0 and t()> 0 for =Dt.Thus, ifXis an optimal strategy, we necessarily have t = Dt, dt-a.e. From (58), we get T

t

(u)(h1P,u)(t)

(u u)f(h1P,u(t)) +uh1P,u(t)f(h1P,u(t))

du+(T)f(t)

dt = 0,

and thusdt = 0since(h1P,u)

andx(uu)f(x)+ uxf(x)are positive functions by assumption. We getthatt= , whereis the solution of (38). In particular, we haveX0 = (0)F(D0+) = (0)F(h

1P,0()) =

X0 and thenX=X. This gives the uniqueness of the optimal strategy. Last, 0 has the same sign as

sinceand have the same sign. Proof of Corollary 2.6: By Assumption2.2 we have t

t > 0, xf

(x)

0 and xx(

xf(x)f(x) )

0, which

gives:

hP,t(x) =

t

2 + xf

(x)f(x)

t

t

1 + xf

(x)f(x)

t

2txx(xf

(x)f(x) )

t

1 + xf(x)

f(x)

t

2 >0.Also, we have sgn(x)hP,t(x) |x|, and hP,t is thus bijective on R. We deduce that sgn(x)h1P,t(x) |x|,which gives that the last trade Thas the same sign as .

Let us assume moreover that (30) holds. Let t = (t)f(t)t

hP,t

(t)1+

tf(t)

f(t)

tt

2 >0. Then,

t = t

tt tt

2t+t

1 +

tf(t)

f(t) t

t

2 +

tf(t)

f(t) t

t

tx( xf

(x)

f(x) )|x=t

t

tt tt2t

+t

1 tt

2 t

t

by Assumption2.2.

= t

2t t

t

2 t t2t t

+t

t t2t t

0 by (30).

References

[1] Aurlien Alfonsi, Antje Fruth, and Alexander Schied. Constrained portfolio liquidation in a limit order

book model. InAdvances in mathematics of finance, volume 83 ofBanach Center Publ., pages 925.Polish Acad. Sci. Inst. Math., Warsaw, 2008.

[2] Aurlien Alfonsi, Antje Fruth, and Alexander Schied. Optimal execution strategies in limit order bookswith general shape functions. Quant. Finance, 10(2):143157, 2010.

[3] Aurlien Alfonsi and Alexander Schied. Optimal trade execution and absence of price manipulations inlimit order book models. SIAM J. Financial Math., 1:490522, 2010.

24


25/25

[4] Aurlien Alfonsi, Alexander Schied, and Alla Slynko. Order Book Resilience, Price Manipulation, andthe Positive Portfolio Problem. SSRN eLibrary, 2011.

[5] Robert Almgren and Neil Chriss. Optimal execution of portfolio transactions. Journal of Risk, 3:539,2000.

[6] Dimitris Bertsimas and Andrew Lo. Optimal control of execution costs. Journal of Financial Markets,

1:150, 1998.

[7] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of PoliticalEconomy, 81(3):pp. 637654, 1973.

[8] Umut etin, Robert A. Jarrow, and Philip Protter. Liquidity risk and arbitrage pricing theory. FinanceStoch., 8(3):311341, 2004.

[9] A. Fruth, T. Schneborn, and M. Urusov. Optimal trade execution and price manipulation in orderbooks with time-varying liquidity. Working Paper Series, 2011.

[10] Jim Gatheral. No-dynamic-arbitrage and market impact. Quant. Finance, 10(7):749759, 2010.

[11] Jim Gatheral, Alexander Schied, and Alla Slynko. Transient linear price impact and fredholm integral

equations. Mathematical Finance, pages nono, 2011.[12] Gur Huberman and Werner Stanzl. Price manipulation and quasi-arbitrage. Econometrica, 72(4):1247

1275, 2004.

[13] A. Obizhaeva and J. Wang. Optimal Trading Strategy and Supply/Demand Dynamics. Working PaperSeries, 2005.

[14] Marc Potters and Jean-Philippe Bouchaud. More statistical properties of order books and price im-pact. Physica A: Statistical Mechanics and its Applications, 324(1-2):133140, 2003. Proceedings of theInternational Econophysics Conference.

[15] Silviu Predoiu, Gennady Shaikhet, and Steven Shreve. Optimal execution in a general one-sided limit-order book. SIAM J. Financial Math., 2:183212, 2011.

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optimal execution and price manipulations in time-varying limit order books

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