Page 1

Optimal execution strategies

in limit order books

with general shape functions

Aur´ elien Alfonsi∗

CERMICS, projet MATHFI

Ecole Nationale des Ponts et Chauss´ ees

6-8 avenue Blaise Pascal

Cit´ e Descartes, Champs sur Marne

77455 Marne-la-vall´ ee, France

alfonsi@cermics.enpc.fr

Antje Fruth

Quantitative Products Laboratory

Alexanderstr. 5

10178 Berlin, Germany

fruth@math.tu-berlin.de

Alexander Schied∗

Department of Mathematics, MA 7-4

TU Berlin

Strasse des 17. Juni 136

10623 Berlin, Germany

schied@math.tu-berlin.de

To appear in Quantitative Finance

Submitted September 3, 2007, accepted July 24, 2008

This version: November 20, 2009

Abstract: We consider optimal execution strategies for block market orders placed in a limit

order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005)

but allow for a general shape of the LOB defined via a given density function. Thus, we can allow

for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We

distinguish two possibilities for modeling the resilience of the LOB after a large market order:

the exponential recovery of the number of limit orders, i.e., of the volume of the LOB, or the

exponential recovery of the bid-ask spread. We consider both of these resilience modes and, in

each case, derive explicit optimal execution strategies in discrete time. Applying our results to

a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a

risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang

(2005). We also provide some evidence for the robustness of optimal strategies with respect to

the choice of the shape function and the resilience-type.

∗Supported by Deutsche Forschungsgemeinschaft through the Research Center Matheon “Mathematics for

key technologies” (FZT 86).

1

hal-00166969, version 3 - 3 Feb 2010

Page 2

1 Introduction.

A common problem for stock traders consists in unwinding large block orders of shares, which

can comprise up to twenty percent of the daily traded volume of shares. Orders of this size

create significant impact on the asset price and, to reduce the overall market impact, it is

necessary to split them into smaller orders that are subsequently placed throughout a certain

time interval. The question at hand is thus to allocate an optimal proportion of the entire order

to each individual placement such that the overall price impact is minimized.

Problems of this type were investigated by Bertsimas and Lo [8], Almgren and Chriss [3, 4],

Almgren and Lorenz [5], Obizhaeva and Wang [16], and Schied and Sch¨ oneborn [18, 19] to

mention only a few. For extensions to situations with several competing traders, see [11], [12],

[20], and the references therein.

The mathematical formulation of the corresponding optimization problem relies first of all

on specifying a stock price model that takes into account the often nonlinear feedback effects

resulting from the placement of large orders by a ‘large trader’. In the majority of models

in the literature, such orders affect the stock price in the following two ways. A first part of

the price impact is permanent and forever pushes the price in a certain direction (upward for

buy orders, downward for sell orders). The second part, which is usually called the temporary

impact, has no duration and only instantaneously affects the trade that has triggered it. It

is therefore equivalent to a (possibly nonlinear) penalization by transaction costs. Models of

this type underlie the above-mentioned papers [8], [3], [4], [5], [11], [12], and [20]. Also the

market impact models described in Bank and Baum [7], Cetin et al. [13], Frey [14], and Frey

and Patie [15] fall into that category. While most of these models start with the dynamics of

the asset price process as a given fundamental, Obizhaeva and Wang [16] recently proposed

a market impact model that derives its dynamics from an underlying model of a limit order

book (LOB). In this model, the ask part of the LOB consists of a uniform distribution of shares

offered at prices higher than the current best ask price. When the large trader is not active,

the mid price of the LOB fluctuates according to the actions of noise traders, and the bid-ask

spread remains constant. A buy market order of the large trader, however, consumes a block

of shares located immediately to the right of the best ask and thus increase the ask price by a

linear proportion of the size of the order. In addition, the LOB will recover from the impact of

the buy order, i.e., it will show a certain resilience. The resulting price impact will neither be

instantaneous nor entirely permanent but will decay on an exponential scale.

The model from [16] is quite close to descriptions of price impact on LOBs found in empirical

studies such as Biais et al. [9], Potters and Bouchaud [17], Bouchaud et al. [10], and Weber

and Rosenow [21]. In particular, the existence of a strong resilience effect, which stems from

the placement of new limit orders close to the bid-ask spread, seems to be a well established

fact, although its quantitative features seem to be the subject of an ongoing discussion.

In this paper, we will pick up the LOB-based market impact model from [16] and generalize

it by allowing for a nonuniform price distribution of shares within the LOB. The resulting

LOB shape which is nonconstant in the price conforms to empirical observations made in

[9, 17, 10, 21]. It also leads completely naturally to a nonlinear price impact of market orders

as found in an empirical study by Almgren et al. [6]; see also Almgren [2] and the references

therein. In this generalized model, we will also consider the following two distinct possibilities

for modeling the resilience of the LOB after a large market order: the exponential recovery

2

hal-00166969, version 3 - 3 Feb 2010

Page 3

of the number of limit orders, i.e., of the volume of the LOB (Model 1), or the exponential

recovery of the bid-ask spread (Model 2). While one can imagine also other possibilities, we

will focus on these two obvious resilience modes. Note that we assume the LOB shape to be

constant in time. Having a time-varying LOB shape will be an area of ongoing research.

We do not have a classical permanent price impact in our model for the following reasons:

Adding classical permanent impact, which is proportional to the volume traded, would be

somewhat artificial in our model. In addition, this would not change optimal strategies as the

optimization problem will be exactly the same as without permanent impact. What one would

want to have instead is a permanent impact with a sensible meaning in the LOB context. But

this would bring substantial difficulties in our derivation of optimal strategies.

After introducing the generalized LOB with its two resilience modes, we consider the prob-

lem of optimally executing a buy order for X0shares within a certain time frame [0,T]. The

focus on buy orders is for the simplicity of the presentation only, completely analogous results

hold for sell orders as well. While most other papers, including [16], focus on optimization

within the class of deterministic strategies, we will here allow for dynamic updating of trad-

ing strategies, that is, we optimize over the larger class of adapted strategies. We will also

allow for intermediate sell orders in our strategies. Our main results, Theorem 4.1 and Theo-

rem 5.1, will provide explicit solutions of this problem in Model 1 and Model 2, respectively.

Applying our results to a block-shaped LOB, we obtain a new closed-form representation for

the corresponding optimal strategy, which explicitly solves the recursive scheme given in [16].

Looking at several examples, we will also find some evidence for the robustness of the optimal

strategy. That is the optimal strategies are qualitatively and quantitatively rather insensitive

with respect to the choice of the LOB shape. In practice, this means that we can use them

even though the LOB is not perfectly calibrated and has a small evolution during the execution

strategy.

The model we are using here is time homogeneous: the resilience rate is constant and trading

times are equally spaced. By using the techniques introduced in our subsequent paper [1], it

is possible to relax these assumptions and to allow for time inhomogeneities and also for linear

constraints, at least in block-shaped models.

The method we use in our proofs is different from the approach used in [16]. Instead of

using dynamic programming techniques, we will first reduce the model of a full LOB with

nontrivial bid-ask spreads to a simplified model, for which the bid-ask spreads have collapsed

but the optimization problem is equivalent. The minimization of the simplified cost functional

is then reduced to the minimization of certain functions that are defined on an affine space.

This latter minimization is then carried out by means of the Lagrange multiplier method and

explicit calculations.

The paper is organized as follows. In Section 2, we explain the two market impact models

that we derive from the generalized LOB model with different resilience modes. In Section 3, we

set up the resulting optimization problem. The main results for Models 1 and 2 are presented

in the respective Sections 4 and 5. In Section 6, we consider the special case of a uniform

distribution of shares in the LOB as considered in [16]. In particular, we provide our new

explicit formula for the optimal strategy in a block-shaped LOB as obtained in [16]. Section 7

contains numerical and theoretical studies of the optimization problem for various nonconstant

shape functions. The proofs of our main results are given in the remaining Sections A through D.

More precisely, in Section A we reduce the optimization problem for our two-sided LOB models

3

hal-00166969, version 3 - 3 Feb 2010

Page 4

to the optimization over deterministic strategies within a simplified model with a collapsed

bid-ask spread. The derivations of the explicit forms of the optimal strategies in Models 1 and

2 are carried out in the respective Sections B and C. In Section D we prove the results for

block-shaped LOBs from Section 6.

2 Two market impact models with resilience.

In this section, we aim at modeling the dynamics of a LOB that is exposed to repeated market

orders by a large trader. The overall goal of the large trader will be to purchase a large amount

X0> 0 of shares within a certain time period [0,T]. Hence, emphasis is on buy orders, and we

concentrate first on the upper part of the LOB, which consists of shares offered at various ask

prices. The lowest ask price at which shares are offered is called the best ask price.

Suppose first that the large trader is not active, so that the dynamics of the limit order

book are determined by the actions of noise traders only. We assume that the corresponding

unaffected best ask price A0is a martingale on a given filtered probability space (Ω,(Ft),F,P)

and satisfies A0

This assumption includes in particular the case in which A0is a

Bachelier model, i.e., A0

We emphasize, however, that we can take any martingale and hence use, e.g., a geometric

Brownian motion, which avoids the counterintuitive negative prices of the Bachelier model.

Moreover, we can allow for jumps in the dynamics of A0so as to model the trading activities

of other large traders in the market. In our context of a risk-neutral investor minimizing the

expected liquidation cost, the optimal strategies will turn out to be deterministic, due to the

described martingale assumption.

Above the unaffected best ask price A0

available shares in the LOB: the number of shares offered at price A0

for a continuous density function f : R −→]0,∞[. We will say that f is the shape function of

the LOB. The choice of a constant shape function corresponds to the block-shaped LOB model

of Obizhaeva and Wang [16].

The shape function determines the impact of a market order placed by our large trader.

Suppose for instance that the large trader places a buy market order for x0> 0 shares at time

t = 0. This market order will consume all shares located at prices between A0and A0+ DA

where DA

?DA

Consequently, the ask price will be shifted up from A0to

0= A0.

t= A0+ σWtfor an (Ft)-Brownian motion W, as considered in [16].

t, we assume a continuous ask price distribution for

t+ x is given by f(x)dx

0+,

0+is determined by

0+

0

f(x)dx = x0.

A0+:= A0+ DA

0+;

see Figure 1 for an illustration.

Let us denote by Atthe actual ask price at time t, i.e., the ask price after taking the price

impact of previous buy orders of the large trader into account, and let us denote by

DA

t:= At− A0

t

4

hal-00166969, version 3 - 3 Feb 2010

Page 5

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

aaaaaaaa

???? ?? ????????

?????? ?? ?????

? ???

???

?

??

?

?

?

??

?

??

?

?

?

??

?

??

?

?

Figure 1: The impact of a buy market order of x0shares .

the extra spread caused by the actions of the large trader. Another buy market order of xt> 0

shares will now consume all the shares offered at prices between Atand

At+:= At+ DA

t+− DA

t= A0

t+ DA

t+,

where DA

t+is determined by the condition

?DA

t+

DA

t

f(x)dx = xt. (1)

Thus, the process DAcaptures the impact of market orders on the current best ask price.

Clearly, the price impact DA

f is constant between DA

functions; see, e.g., Almgren [2] and Almgren et al. [6] for a discussion.

Another important quantity is the process

t+− DA

tand DA

t will be a nonlinear function of the order size xt unless

t+. Hence, our model includes the case of nonlinear impact

EA

t=

?DA

t

0

f(x)dx, (2)

of the number of shares ‘already eaten up’ at time t. It quantifies the impact of the large trader

on the volume of the LOB. By introducing the antiderivative

?z

of f, the relation (2) can also be expressed as

F(z) =

0

f(x)dx (3)

EA

t= F(DA

t) andDA

t= F−1(EA

t),(4)

where we have used our assumption that f is strictly positive to obtain the second identity.

The relation (1) is equivalent to

EA

t+= EA

t+ xt. (5)

5

hal-00166969, version 3 - 3 Feb 2010

Page 6

We still need to specify how DAand, equivalently, EAevolve when the large trader is

inactive in between market orders. It is a well established empirical fact that order books

exhibit a certain resilience as to the price impact of a large buy market orders, i.e., after the

initial impact the best ask price reverts back to its previous position; cf. Biais et al. [9], Potters

and Bouchaud [17], Bouchaud et al. [10], and Weber and Rosenow [21] for empirical studies.

That is, at least a part of the price impact will only be temporary. For modeling this resilience,

we follow Obizhaeva and Wang [16] in proposing an exponential recovery of the LOB. While in

the case of a block-shaped LOB as considered in [16] the respective assumptions of exponential

recovery for DAand for EAcoincide, they provide two distinct possibilities for the case of a

general shape function. Since either of them appears to be plausible, we will discuss them both

in the sequel. More precisely, we will consider the following two models for the resilience of the

market impact:

Model 1: The volume of the order book recovers exponentially, i.e., E evolves according to

EA

t+s= e−ρsEA

t

(6)

if the large investor is inactive during the time interval [t,t + s[.

Model 2: The extra spread DA

tdecays exponentially, i.e.,

DA

t+s= e−ρsDA

t

(7)

if the large investor is inactive during the time interval [t,t + s[.

Here the resilience speed ρ is a positive constant, which for commonly traded blue chip

shares will often be calibrated such that the half-life time of the exponential decay is in the

order of a few minutes; see, e.g., [17, 10, 21]. Note that the dynamics of both DAand EAare

now completely specified in either model.

Up to now, we have only described the effect of buy orders on the upper half of the LOB.

Since the overall goal of the larger trader is to buy X0> 0 shares up to time T, a restriction

to buy orders would seem to be reasonable. However, we do not wish to exclude the a priori

possibility that, under certain market conditions, it could be beneficial to also sell some shares

and to buy them back at a later point in time. To this end, we also need to model the impact

of sell market orders on the lower part of the LOB, which consists of a certain number of bids

for shares at each price below the best bid price. As for ask prices, we will distinguish between

an unaffected best bid price, B0

of previous sell orders of the large trader is taken into account. All we assume on the dynamics

of B0is

B0

t

at all times t.

The distribution of bids below B0

the domain ] − ∞,0]. More precisely, for x < 0, the number of bids at price B0

f(x)dx. The quantity

DB

t, and the actual best bid price, Bt, for which the price impact

t≤ A0

tis modeled by the restriction of the shape function f to

(8)

t+ x is equal to

t:= Bt− B0

t,

which usually will be negative, is called the extra spread in the bid price distribution. A sell

market order of xt < 0 shares placed at time t will consume all the shares offered at prices

between Btand

Bt+:= Bt+ DB

t+− DB

t= B0

t+ DB

t+,

6

hal-00166969, version 3 - 3 Feb 2010

Page 7

where DB

t+is determined by the condition

xt=

?DB

t+

DB

t

f(x)dx = F(DB

t+) − F(DB

t) = EB

t+− EB

t, (9)

for EB

trader is inactive during the time interval [t,t+ s[, then the processes DBand EBbehave just

as their counterparts DAand EA, i.e.,

s:= F(DB

s). Note that F is defined via (3) also for negative arguments. If the large

EB

DB

t+s= e−ρsEB

t+s= e−ρsDB

t

in Model 1,

t

in Model 2.

(10)

3 The cost minimization problem.

When placing a single buy market order of size xt≥ 0 at time t, the large trader will purchase

f(x)dx shares at price A0

buy market order amounts to

?DA

For a sell market order xt≤ 0, we have

?DB

In practice, very large orders are often split into a number of consecutive market orders to

reduce the overall price impact. Hence, the question at hand is to determine the size of the

individual orders so as to minimize a cost criterion. So let us assume that the large trader needs

to buy a total of X0> 0 shares until time T and that trading can occur at N + 1 equidistant

times tn = nτ for n = 0,...,N and τ := T/N. An admissible strategy will be a sequence

ξ = (ξ0,ξ1,...,ξN) of random variables such that

•?N

• each ξnis measurable with respect to Ftn,

• each ξnis bounded from below.

The quantity ξncorresponds to the size of the market order placed at time tn. Note that we

do not a priori require ξnto be positive, i.e., we also allow for intermediate sell orders, but we

assume that there is some lower bound on sell orders.

The average cost C(ξ) of an admissible strategy ξ is defined as the expected value of the

total costs incurred by the consecutive market orders:

t+ x, with x ranging from DA

tto DA

t+. Hence, the total cost of the

πt(xt) :=

t+

DA

t

(A0

t+ x)f(x)dx = A0

txt+

?DA

t+

DA

t

xf(x)dx.(11)

πt(xt) := B0

txt+

t+

DB

t

xf(x)dx. (12)

n=0ξn= X0,

C(ξ) = E

?

N

?

7

n=0

πtn(ξn)

?

. (13)

hal-00166969, version 3 - 3 Feb 2010

Page 8

Our goal in this paper consists in finding admissible strategies that minimize the average cost

within the class of all admissible strategies. For the clarity of the exposition, we decided no to

treat the case of a risk averse investor. We suppose that the introduction of risk aversion will

have a similar effect as in [16].

Note that the value of C(ξ) depends on whether we choose Model 1 or Model 2, and it

will turn out that also the quantitative—though not the qualitative—features of the optimal

strategies will be slightly model-dependent.

Before turning to the statements of our results, let us introduce the following standing

assumption for our further analysis: the function F is supposed to be unbounded in the sense

that

lim

x↑∞F(x) = ∞ and lim

x↓−∞F(x) = −∞. (14)

This assumption of unlimited order book depth is of course an idealization of reality and is for

convenience only. It should not make a difference, however, as soon as the depth of the real

LOB is big enough to accommodate every market order of our optimal strategy.

4 Main theorem for Model 1.

We will now consider the minimization of the cost functional C(ξ) in Model 1, in which we

assume an exponential recovery of the LOB volume; cf. (6).

Theorem 4.1 (Optimal strategy in Model 1).

Suppose that the function h1: R → R+with

h1(y) := F−1(y) − e−ρτF−1(e−ρτy)

is one-to-one. Then there exists a unique optimal strategy ξ(1)= (ξ(1)

market order ξ(1)

0

is the unique solution of the equation

0,...,ξ(1)

N). The initial

F−1?

X0− Nξ(1)

0

?1 − e−ρτ??

=h1(ξ(1)

1 − e−ρτ,

0)

(15)

the intermediate orders are given by

ξ(1)

1

= ··· = ξ(1)

N−1= ξ(1)

0

?1 − e−ρτ?, (16)

and the final order is determined by

ξ(1)

N= X0− ξ(1)

0 − (N − 1)ξ(1)

0

?1 − e−ρτ?.

In particular, the optimal strategy is deterministic. Moreover, it consists only of nontrivial buy

orders, i.e., ξ(1)

n > 0 for all n.

Some remarks on this result are in order. First, the optimal strategy ξ(1)consists only of

buy orders and so the bid price remains unaffected, i.e., we have EB

t ≡ 0 ≡ DB

t. It follows

8

hal-00166969, version 3 - 3 Feb 2010

Page 9

moreover that the process E := EAis recursively given by the following Model 1 dynamics:

E0 = 0,

Etn+ = Etn+ ξ(1)

= e−ρτEtk+= e−ρτ(Etk+ ξ(1)

n,n = 0,...,N, (17)

Etk+1

k),k = 0,...,N − 1.

Hence, by (15) and (16),

Etn+= ξ(1)

0

and Etn+1= e−ρτξ(1)

0

for n = 0,...,N − 1. (18)

That is, once ξ(1)

market orders that consume exactly that amount of shares by which the LOB has recovered

since the preceding market order, due to the resilience effect. At the terminal time tN= T, all

remaining shares are bought. In the case of a block-shaped LOB, this qualitative pattern was

already observed by Obizhaeva and Wang [16]. Our Theorem 4.1 now shows that this optimality

pattern is actually independent of the LOB shape, thus indicating a certain robustness of

optimal strategies.

0

has been determined via (15), the optimal strategy consists in a sequence of

Remark 4.2 According to (4) and (18), the extra spread D := DAof the optimal strategy ξ(1)

satisfies

Dtn+= F−1(Etn+) = F−1(ξ(1)

0).

For n = N we moreover have that

DtN+ = F−1(EtN+) = F−1?

EtN+ ξ(1)

N

?

= F−1?

ξ(1)

0e−ρτ+ X0− ξ(1)

X0− Nξ(1)

0 − (N − 1)ξ(1)

?1 − e−ρτ??

0

?1 − e−ρτ??

= F−1?

0

.

Hence, the left-hand side of (15) is equal to DtN+.

We now comment on the conditions in Theorem 4.1.

Remark 4.3 (When is h1one-to-one?) The function h1is continuous with h1(0) = 0 and h1(y) >

0 for y > 0. Hence, h1is one-to-one if and only if h1is strictly increasing. We want to consider

when this is the case. To this end, note that the condition

h′

1(y) =

1

f(F−1(y))−

e−2ρτ

f(F−1(e−ρτy))> 0

is equivalent to

ℓ(y) := f(F−1(e−ρτy)) − e−2ρτf(F−1(y)) > 0. (19)

That is, the function h1will be one-to-one if, for instance, the shape function f is decreasing

for y > 0 and increasing for y < 0. In fact, it has been observed in the empirical studies

[9, 17, 10, 21] that average shapes of typical order books have a maximum at or close to the best

quotes and then decay as a function of the distance to the best quotes, which would conform to

our assumption.

9

hal-00166969, version 3 - 3 Feb 2010

Page 10

Remark 4.4 (Continuous-time limit of the optimal strategy). One can also investigate the

asymptotic behavior of the optimal strategy when the number N of trades in ]0,T] tends to

infinity. It is not difficult to see that h1/(1 − e−ρτ) converges pointwise to

h∞

1(y) := F−1(y) +

y

f(F−1(y)).

Observe also that N(1 − e−ρτ) → ρT. Since for any N we have ξ(1)

subsequence that converges and its limit is then necessarily solution of the equation

0

∈]0,X0[, we can extract a

F−1(X0− ρTy) = h∞

1(y).

If this equation has a unique solution ξ(1),∞

to ξ(1),∞

0

when N −→ ∞. This is the case, for example, if h∞

especially when f is decreasing. In that case, Nξ(1)

X0− ξ(1),∞

initial block order of ξ(1),∞

0

shares at time 0, continuous buying at the constant rate ρξ(1),∞

during ]0,T[, and a final block order of ξ(1),∞

T

shares at time T.

0

we deduce that the optimal initial trade converges

is strictly increasing and

converges to ρTξ(1),∞

0

1

1

and ξ(1)

Nto ξ(1),∞

T

:=

0

(1 + ρT). Thus, in the continuous-time limit, the optimal strategy consists in an

0

5 Main theorem for Model 2.

We will now consider the minimization of the cost functional

C(ξ) = E

?

N

?

n=0

πtn(ξn)

?

in Model 2, where we assume an exponential recovery of the extra spread; cf. (7).

Theorem 5.1 (Optimal strategy in Model 2).

Suppose that the function h2: R → R with

h2(x) := xf(x) − e−2ρτf(e−ρτx)

f(x) − e−ρτf(e−ρτx)

is one-to-one and that the shape function satisfies

lim

|x|→∞x2

inf

z∈[e−ρτx,x]f(z) = ∞.(20)

Then there exists a unique optimal strategy ξ(2)= (ξ(2)

is the unique solution of the equation

F−1?

the intermediate orders are given by

0,...,ξ(2)

N). The initial market order ξ(2)

0

X0− N?ξ(2)

0 − F?e−ρτF−1(ξ(2)

0)???

= h2

?F−1(ξ(2)

0)?, (21)

ξ(2)

1

= ··· = ξ(2)

N−1= ξ(2)

0 − F?e−ρτF−1(ξ(2)

10

0)?,(22)

hal-00166969, version 3 - 3 Feb 2010

Page 11

and the final order is determined by

ξ(2)

N= X0− Nξ(2)

0 + (N − 1)F?e−ρτF−1(ξ(2)

0)?.

In particular, the optimal strategy is deterministic. Moreover, it consists only of nontrivial buy

orders, i.e., ξ(2)

n > 0 for all n.

Since the optimal strategy ξ(2)consists only of buy orders, the processes DBand EBvanish,

and D := DAis given by

D0 = 0,

Dtn+ = F−1?ξ(2)

Dtk+1

n + F (Dtn)?,n = 0,...,N (23)

= e−ρτDtk+,k = 0,...,N − 1.

Hence, induction shows that

Dtn+= F−1(ξ(2)

0) andDtn+1= e−ρτF−1(ξ(2)

0) for n = 0,...,N − 1.

By (4), the process E := EAsatisfies

Etn+= ξ(2)

0

andEtn+1= F?e−ρτF−1(ξ(2)

0)?

for n = 0,...,N − 1.

has been determined via (15), theThis is very similar to our result (18) in Model 1: once ξ(1)

optimal strategy consists in a sequence of market orders that consume exactly that amount

of shares by which the LOB has recovered since the preceding market order. At the terminal

time tN= T, all remaining shares are bought. The only differences are in the size of the initial

market order and in the mode of recovery. This qualitative similarity between the optimal

strategies in Models 1 and 2 again confirms our observation of the robustness of the optimal

strategy.

0

Remark 5.2 At the terminal time tN= T, the extra spread is given by

DtN+ = F−1(EtN+) = F−1?EtN+ ξ(2)

N

?

= F−1?

X0− N?ξ(2)

0 − F?e−ρτF−1(ξ(2)

0)???

,

and this expression coincides with the left-hand side in (21).

Let us now comment on the conditions assumed in Theorem 5.1. To this end, we first

introduce the function

?F(z) :=

Remark 5.3 If?F is convex then condition (20) in Theorem 5.1 is satisfied. This fact admits

x2

inf

?z

0

xf(x)dx.(24)

the following short proof. Take x∗∈ [e−ρτx,x] realizing the infimum of f in [e−ρτx,x]. Then

z∈[e−ρτx,x]f(z) = x2f(x∗) ≥ x∗(x∗f(x∗)).(25)

Due to the convexity of?F, its derivative?F′(x) = xf(x) is increasing. It is also nonzero iff

11

x ?= 0. Therefore the right-hand side of (25) tends to infinity for |x| → ∞.

hal-00166969, version 3 - 3 Feb 2010

Page 12

However, the convexity of?F is not necessary for condition (20) as is illustrated by the

Example 5.4 Let us construct a shape function for which (20) is satisfied even though?F need

from zero. Then let

?b(1)

following simple example.

not be convex. To this end, take any continuous function b : R →]0,∞[ that is bounded away

f(x) :=

|x| ≤ 1

|x| > 1.

b(x)

√

|x|

This shape function clearly satisfies condition (20). Taking for example b(x) = 1+εcos(x) with

0 < ε < 1, however, gives a nonconvex function?F. Moreover, by choosing ε small enough, we

can obtain h′

2(x) > 0 so that the shape function f satisfies the assumptions of Theorem 5.1.

We now comment on the condition that h2is one-to-one. The following example shows that

this is indeed a nontrivial assumption.

Example 5.5 We now provide an example of a shape function f for which the corresponding

function h2is not one-to-one. First note that h2(0) = 0 and

lim

ǫ↓0

h2(ǫ) − h2(0)

ǫ

=1 − e−2ρτ

1 − e−ρτ> 0. (26)

Therefore and since h2 is continuous, it cannot be one-to-one if we can find x∗> 0 such

that h2(x∗) < 0. To this end, we assume that there exist n ∈ {2,3,...} such that e−ρτ=1

take

see Figure 2. Furthermore, we define x∗:= 1 to obtain

nand

f(x) :=

(n + 1)

(n + 1) −

1

x ∈?0,1

x ∈ (1,∞);

n

?

n2

n−1

?x −1

n

?

x ∈?1

n,1?

h2(x∗) =n2− (n + 1)

−n

< 0.

The intuition why Theorem 4.1 can be applied to this LOB shape (f is decreasing), but

Theorem 5.1 cannot be used, is the following: For the first trade ξ(2)

Dtn+1= e−ρτF−1(ξ(2)

f(x) is low for x ≥ 1. But this ξ(2)

in Model 1 because there the resilience is proportional to the volume consumed by the large

investor.

0

from (21) we might get

0) ≥ 1, i.e. there are only few new shares from the resilience effect since

0

would not be optimal1. We cannot have this phenomenon

1Take e.g. n = 2 and e−ρτ= 1/2. Then for X0 = N +9

ξ(2)

1

= ... = ξ(2)

ξ

1

= ... = ξ

2we get from (21) ξ(2)

0

=

7

2, Dtn+1= 1 and

N−1= 1, ξ(2)

(2)

N−1= 1, ξ

N= 2. The corresponding cost are higher than for the alternative strategy ξ

(2)

N= 3.

(2)

0

=5

2,

(2)

12

hal-00166969, version 3 - 3 Feb 2010

Page 13

n + 1

0

f(x)

1

1

n

x∗= 1

x

Figure 2: A shape function f for which the function h2is not one-to-one.

Remark 5.6 (Continuous-time limit of the optimal strategy). As in Remark 4.4, we can

study the asymptotic behavior of the optimal strategy as the number N of trades in ]0,T[ tends

to infinity. First, we can check that h2converges pointwise to

h∞

2(x) := x(1 +

f(x)

f(x) + xf′(x)),

and that N(y − F(e−ρτF−1(y))) tends to ρTF−1(y)f(F−1(y)), provided that f is continuously

differentiable. Now, suppose that the equation

F−1(X0− ρTF−1(y)f(F−1(y))) = h∞

2(F−1(y))

has a unique solution on ]0,X0[, which we will call ξ(2),∞

one possible limit for a subsequence of ξ(2)

Nξ(2)

1

converges to ρTF−1(ξ(2),∞

0

0

. We can check that ξ(2),∞

0

is the only

0, and it is therefore its limit. We can then show that

)f(F−1(ξ(2),∞

0

)) and ξ(2)

Nto

ξ(2),∞

T

:= X0− ξ(2),∞

0

− ρTF−1(ξ(2),∞

0

)f(F−1(ξ(2),∞

0

)).

Thus, in the continuous-time limit, the optimal strategy consists in an initial block order of

ξ(2),∞

0

shares at time 0, continuous buying at the constant rate ρF−1(ξ(2),∞

]0,T[, and a final block order of ξ(2),∞

T

shares at time T.

0

)f(F−1(ξ(2),∞

0

)) during

6Closed form solution for block-shaped LOBs and ad-

ditional permanent impact.

In this first example section, we consider a block-shaped LOB corresponding to a constant

shape function f(x) ≡ q for some q > 0. In this case, there is no difference between Models 1

and 2. Apart from our more general dynamics for A0, the main difference to the market impact

model introduced by Obizhaeva and Wang [16] is that, for the moment, we do not consider a

permanent impact of market orders. In Corollary 6.4, we will see, however, that our results

yield a closed-form solution even in the case of nonvanishing permanent impact.

By applying either Theorem 4.1 or Theorem 5.1 we obtain the following Corollary.

13

hal-00166969, version 3 - 3 Feb 2010

Page 14

Corollary 6.1 (Closed-form solution for block-shaped LOB).

In a block-shaped LOB, the unique optimal strategy ξ∗is

ξ∗

0= ξ∗

N=

X0

(N − 1)(1 − e−ρτ) + 2

and ξ∗

1= ··· = ξ∗

N−1=X0− 2ξ∗

0

N − 1

. (27)

The preceding result extends [16, Proposition 1] in several aspects. First, we do not focus on

the Bachelier model but admit arbitrary martingale dynamics for our unaffected best ask price

A0. Second, only static, deterministic buy order strategies are considered in [16], while we here

allow our admissible strategies to be adapted and to include sell orders. Since, a posteriori,

our optimal strategy turns out to be deterministic and positive, it is clear that it must coincide

with the optimal strategy from [16, Proposition 1]. Our strategy (27) therefore also provides an

explicit closed-form solution of the recursive scheme obtained in [16]. We recall this recursive

scheme in (31) below.

On the other hand, Obizhaewa and Wang [16] allow for an additional permanent impact of

market orders. Intuitively, in a block-shaped LOB with f ≡ q > 0, the permanent impact of

a market order xtmeans that only a certain part of the impact of xtdecays to zero, while the

remaining part remains forever present in the LOB. More precisely, the impact of an admissible

buy order strategy ξ on the extra spread DAis given by the dynamics

?

where λ < 1/q is a constant quantifying the permanent impact and

DA

t= λ

tk<t

ξk+

?

tk<t

κe−ρ(t−tk)ξk, (28)

κ :=1

q− λ(29)

is the proportion of the temporary impact. Note that, for λ = 0, we get back our dynamics (6)

and (7), due to the fact that we consider a block-shaped LOB. It will be convenient to introduce

the process Xtof the still outstanding number of shares at time t when using an admissible

strategy:

Xt:= X0−

?

tk<t

ξk. (30)

We can now state the result by Obizhaeva and Wang.

Proposition 6.2 [16, Proposition 1] In a block-shaped LOB with permanent impact λ, the

optimal strategy ξOWin the class of deterministic strategies is determined by the forward scheme

ξOW

n

=

1

2δn+1[ǫn+1Xtn− φn+1Dtn],

= XT,

n = 0,...,N − 1, (31)

ξOW

N

where δn, ǫnand φnare defined by the backward scheme

δn :=

?1

2q+ αn− βnκe−ρτ+ γnκ2e−2ρτ?−1

ǫn := λ + 2αn− βnκe−ρτ

φn := 1 − βne−ρτ+ 2γnκe−2ρτ.

(32)

14

hal-00166969, version 3 - 3 Feb 2010

Page 15

with αn, βnand γngiven by

αN=

1

2q− λ and αn = αn+1−1

βN= 1 and βn = βn+1e−ρτ+1

4δn+1ǫ2

n+1,

2δn+1ǫn+1φn+1,

4δn+1φ2

(33)

γN= 0 and γn = γn+1e−2ρτ−1

n+1.

It is a priori clear that for λ = 0 the explicit optimal strategy obtained in Corollary 6.1

must coincide with the strategy ξOWobtained via the recursive scheme (31) in Proposition 6.2.

To cross-check our results with the ones in [16], we will nevertheless provide an explicit and

independent proof of the following proposition. It can be found in Section D.

Proposition 6.3 For λ = 0, the optimal strategy (27) of Corollary 6.1 solves the recursive

scheme (31) in Proposition 6.2.

Let us now extend our results so as to obtain the explicit solution of (31) even with nonvan-

ishing permanent impact. To this end, we note that the optimal strategy ξOW= (ξOW

is obtained in [16] as the unique minimizer of the cost functional

0

,...,ξOW

N )

COW

λ,q: RN+1→ R

defined by

COW

λ,q(x0,...,xN)

?

= A0

N

i=0

xi+λ

2

? N

?

i=0

xi

?2

+ κ

N

?

k=0

?k−1

?

i=0

xie−ρ(k−i)τ?

xk+κ

2

N

?

i=0

x2

i,

where κ is as in (29). Now we just have to observe that

COW

λ,q(x0,...,xN) =λ

2

? N

?

i=0

xi

?2

+ COW

0,κ−1(x0,...,xN).

Therefore, under the constraint?N

fact independent of q. Hence, ξ∗also minimizes COW

proved:

i=0xi= X0, it is equivalent to minimize either COW

λ,qor COW

0,q. But ξ∗is in

λ,q. We have therefore

0,κ−1.

We already know that the optimal strategy ξ∗of Corollary 6.1 minimizes COW

0,κ−1 and in turn COW

Corollary 6.4 The optimal strategy ξ∗of Corollary 6.1 is the unique optimal strategy in any

block-shaped LOB with permanent impact λ < 1/q.

scheme (31).

In particular, it solves the recursive

The last part of the assertion of Corollary 6.4 is remarkable insofar as the recursive scheme (31)

depends on both q and λ whereas the optimal strategy ξ∗does not.

15

hal-00166969, version 3 - 3 Feb 2010

Page 16

0.00.2 0.40.6 0.81.0

5000

10000

15000

20000

x

f(x)

Figure 3: Plots of the power law shape functions for q = 5,000 shares and exponent α =

−2,−1,0,1

real-world shape functions.

2and 1 top down. Please note that these examples do not necessarily correspond to

7 Examples.

In this section, we consider the power law family f : R → R>0with

f(x) =

q

(|x| + 1)α

(34)

as example shape functions. The antiderivative of the shape function and its inverse are

for positive values of x and y. Set F(x) = −F(−|x|) and F−1(y) = −F−1(−|y|) for x,y < 0.

One can easily check that the assumptions of both Theorem 4.1 and Theorem 5.1 are

satisfied for α ≤ 1. It is remarkable that the optimal strategies (Figure 4) vary only slightly

when changing α or the resilience mode. This observation provides further evidence for the

robustness and stability of the optimal strategy, and this time not only on a qualitative but

also on a quantitative level.

From Figure 4 one recognizes some monotonicity properties of the optimal strategies. We

want to give some intutition to understand these. Let us start with Model 1. There the

dynamics of Etdo not depend on the LOB shape, but solely on the strategy. Only the cost

depends on f. We know from the constant LOB case that the optimum strategy is not sensible

to the value of f(ξ(1)

along the different LOB shapes. Moreover, ξ(1)

1

to ξ(1)

0

since it is the number of shares that reappear between two trades. Therefore the optimal

strategy is just a trade-off between ξ(1)

0

and ξ(1)

trade is relatively more (less) expansive compared to the last one. This explains that ξ(1)

for α < 0 and ξ(1)

0

> ξ(1)

LOB case’ (α = 0) where ξ0= ξN.

F(x) =

q log(x + 1)

qx

q

1−α[(x + 1)1−α− 1]

if α = 1

if α = 0

otherwise

F−1(y) =

e

y

q

?

y

q− 1 if α = 1

if α = 0

1 + (1 − α)y

q

?

1

1−α− 1 otherwise

0). This explains why there are few quantitative differences for Model 1

= (1 − a)ξ(1)

0

with a := e−ρτis proportional

N. When f is increasing (decreasing), the first

0

< ξ(1)

N

Nfor α > 0. With ‘relatively’ we mean ’with respect to the constant

16

hal-00166969, version 3 - 3 Feb 2010

Page 17

?2.0

?1.5

?1.0

?0.50.51.0

10000

11000

12000

13000

α

Initial and last trade

?2.0

?1.5

?1.0

?0.5 0.51.0

8400

8600

8800

α

Intermediate trades

Figure 4: The plots show the optimal strategies for varying exponents α. We set X0= 100,000

and q = 5,000 shares, ρ = 20, T = 1 and N = 10. In the left figure we see ξ(1)

thick), ξ(1)

and ξ(2)

0

(dashed and

(thick line)

N(thick line) and ξ(2)

1.

0, ξ(2)

N. The figure on the right hand side shows ξ(1)

1

For Model 2 the dynamics of Et do depend on the shape function, which explains more

substantial variations according to f. Here the main idea is to realize that, for increasing

(decreasing) shape functions, resilience of the volume is stronger (weaker) in comparison to

Model 1. Indeed, we have then x−F(aF−1(x)) ≥ x(1−a) (resp. x−F(aF−1(x)) ≤ x(1−a)).

Therefore ξ(1)

1

< ξ(2)

1

(ξ(1)

1

> ξ(2)

Model 1. These effects are the more pronounced the steeper the LOB shape. Furthermore,

there is the tendency that ξ(2)

0

≈ ξ(2)

suggests ξ(2)

0

< ξ(2)

the number of reappearing shares grows disproportionately in the initial trade which favors the

initial trade being higher than the last trade. These two effects seem to counterbalance each

other.

1) and the discrete trades ξ(2)

0

and ξ(2)

Nare lower (higher) as in

N. On the one hand, the same argument as in Model 1

Nfor increasing f. But on the other hand, for an increasing shape function

Remark 7.1 Taking the special LOB shape f(x) =

q

√

1+µ|x|, q > 0 and µ ≥ 0 we can solve

explicitly the optimal strategy in Model 1 from Theorem 4.1. The optimal initial trade is given

by

ξ(1)

0

=

1 + a + N(1 − a)(1 + (µ/2q)X0)

(µ/2q)(N2(1 − a)2− (1 + a + a2))

?(N + 1 − a(N − 1))2+ (µ/q)X0[N(1 − a2) + (1 + a + a2)(1 + (µ/4q)X0)]

and we can show that it is increasing with respect to the parameter µ that tunes the slope of the

LOB.

−

(µ/2q)(N2(1 − a)2− (1 + a + a2))

,

AReduction to the case of deterministic strategies.

In this section, we prepare for the proofs of Theorems 4.1 and 5.1 by reducing the minimization

of the cost functional

?

C(ξ) = E

N

?

17

n=0

πtn(ξn)

?

hal-00166969, version 3 - 3 Feb 2010

Page 18

with respect to all admissible strategies ξ to the minimization of certain cost functions C(i):

RN+1→ R, where i = 1,2 refers to the model under consideration.

To this end, we introduce simplified versions of the model dynamics by collapsing the bid-

ask spread into a single value. More precisely, for any admissible strategy ξ, we introduce a

new pair of processes D and E that react on both sell and buy orders according to the following

dynamics.

• We have E0= D0= 0 and

Et= F(Dt) andDt= F−1(Et). (35)

• For n = 0,...,N, regardless of the sign of ξn,

Etn+= Etn+ ξn

andDtn+= F−1(ξn+ F (Dtn)). (36)

• For k = 0,...,N − 1,

Etk+1= e−ρτEtk+

Dtk+1= e−ρτDtk+

in Model 1,

in Model 2.

(37)

The values of Etand Dtfor t / ∈ {t0,...,tN} will not be needed in the sequel. Note that E = EA

and D = DAif ξ consists only of buy orders, while E = EBand D = DBif ξ consists only of

sell orders. In general, we will only have

EB

t≤ Et≤ EA

t

andDB

t≤ Dt≤ DA

t. (38)

We now introduce the simplified price of ξnat time tnby

πtn(ξn) := A0

tnξn+

?Dtn+

Dtn

xf(x)dx,(39)

regardless of the sign of ξn. Using (38) and (8), we easily get

πtn(ξn) ≤ πtn(ξn) with equality if ξk≥ 0 for all k ≤ n. (40)

The simplified price functional is defined as

C(ξ) := E

?

N

?

n=0

πtn(ξn)

?

.

We will show that, in Model i ∈ {1,2}, the simplified price functional C has a unique minimizer,

which coincides with the corresponding optimal strategy ξ(i)as described in the respective

theorem. We will also show that ξ(i)consists only of buy orders, so that (40) will yield C(ξ(i)) =

C(ξ(i)). Consequently, ξ(i)must be the unique minimizer of C.

Let us now reduce the minimization of C to the minimization of functionals C(i)defined on

deterministic strategies. To this end, let us use the notation

?

Xt:= X0−

tk<t

ξkfor t ≤ T and XtN+1:= 0. (41)

18

hal-00166969, version 3 - 3 Feb 2010

Page 19

The accumulated simplified price of an admissible strategy ξ is

N

?

n=0

πtn(ξn) =

N

?

n=0

A0

tnξn+

N

?

n=0

?Dtn+

Dtn

xf(x)dx.

Integrating by parts yields

N

?

n=0

A0

tnξn= −

N

?

n=0

A0

tn(Xtn+1− Xtn) = X0A0+

N

?

n=1

Xtn(A0

tn− A0

tn−1). (42)

Since ξ is admissible, Xt is a bounded predictable process. Hence, due to the martingale

property of the unaffected best ask process A0, the expectation of (42) is equal to X0A0.

Next, observe that, in each Model i = 1,2, the simplified extra spread process D evolves de-

terministically once the values ξ0,ξ1(ω),...,ξN(ω) are given. Hence, there exists a deterministic

function C(i): RN+1→ R such that

N

?

It follows that

C(ξ) = A0X0+ E?C(i)(ξ0,...,ξN)?.

We will show in the respective Sections B and C that the functions C(i), i = 1,2, have unique

minima within the set

n=0

?Dtn+

Dtn

xf(x)dx = C(i)(ξ0,...,ξN).(43)

Ξ :=

?

(x0,...,xN) ∈ RN+1??

N

?

n=0

xn= X0

?

,

and that these minima coincide with the values of the optimal strategies ξ(i)as provided in

Theorems 4.1 and 5.1. This concludes the reduction to the case of deterministic strategies. We

will now turn to the minimization of the functions C(i)over Ξ. To simplify the exposition, let

us introduce the following shorthand notation in the sequel:

a := e−ρτ.(44)

B The optimal strategy in Model 1.

In this section, we will minimize the function C(1)of (43) over the set Ξ of all deterministic

strategies and thereby complete the proof of Theorem 4.1. To this end, recall first the definition

of the two processes E and D as given in (35)–(37). Based on their Model 1 dynamics, we will

now obtain a formula of the cost function C(1)of (43) in terms of the functions F and?F. It

G(y) :=?F?F−1(y)?.

will be convenient to introduce also the function

(45)

19

hal-00166969, version 3 - 3 Feb 2010

Page 20

Then we have for any deterministic strategy ξ = (x0,...,xN) ∈ Ξ that

N

?

N

?

N

?

=G(x0) − G(0)

+G(ax0+ x1) − G(ax0)

+G?a2x0+ ax1+ x2

+...

+G?aNx0+ ··· + xN

The derivative of G is

C(1)(x0,...,xN) =

n=0

?Dtn+

??F?F−1(Etn+)?−?F?F−1(Etn)??

?G(Etn+ xn) − G(Etn)?

Dtn

xf(x)dx

=

n=0

=

n=0

(46)

?− G?a2x0+ ax1

?− G?aNx0+ ··· + axN−1

?

(47)

?.

G′(y) =?F′?F−1(y)?(F−1)′(y) = F−1(y)f?F−1(y)?

Hence, G is twice continuously differentiable, positive and convex. The cost function C(1)is

also twice continuously differentiable.

1

f(F−1(y))= F−1(y). (48)

Lemma B.1 We have C(1)(x0,...,xN)−→+∞ for |ξ| → ∞, and therefore there exists a local

minimum of C(1)in Ξ.

Proof: Using (48) and the fact that F−1(yx) is increasing, we get that for all y ∈ R and c ∈ (0,1]

G(y) − G(cy) ≥ (1 − c) · |F−1(cy)| · |y|.

Let us rearrange the sum in (47) in order to use inequality (49). We obtain

(49)

C(1)(x0,...,xN)

= G?aNx0+ aN−1x1+ ··· + xN

+

?− G(0)

?− G?a(anx0+ ··· + xn)??

?− G(0)

??F−1?a(anx0+ ··· + xn)???|anx0+ ··· + xn|.

N−1

?

n=0

?

G?anx0+ ··· + xn

≥ G?aNx0+ aN−1x1+ ··· + xN

+(1 − a)

n=0

N−1

?

Let us denote by T1: RN+1→ RN+1the linear mapping

T1(x0,...,xn) =?x0,ax0+ x1,...,aNx0+ x1aN−1+ ··· + xN

?.

20

hal-00166969, version 3 - 3 Feb 2010

Page 21

It is non trivial and therefore the norm of T1(x0,...,xN) tends to infinity as the norm of its

argument goes to infinity. Because F is unbounded, we know that both G(y) and |F−1(ay)||y|

tend to infinity for |y| → ∞. Let us introduce

H(y) = min(G(y),|F−1(ay)||y|).

Then also H(y)−→ + ∞ for |y| → ∞, and we conclude that

C(1)(x0,...,xN) ≥ (1 − a)H(|T1(x0,...,xN)|∞) − G(0),

where | · |∞denotes the ℓ∞-norm on RN+1. Hence, the assertion follows.

We now consider Equation (15) in Theorem 4.1, which we recall here for the convenience of

the reader:

F−1(X0− Nx0(1 − a)) =h1(x0)

This equation is solved by x0if and only if x0is a zero of the function

ˆh1(y) := h1(y) − (1 − a)F−1?X0− Ny(1 − a)?.

1 − a.

(50)

Lemma B.2 Under the assumptions of the Theorem 4.1,ˆh1has at most one zero x0, which,

if it exists, is necessarily positive.

Proof: It is sufficient to show thatˆh1is strictly increasing. We know that h1(0) = 0, h1(y) > 0

for y > 0, and h1is continuous and one-to-one. Consequently, h1must be strictly increasing

and therefore

ˆh′

f?F−1(X0+ Ny (a − 1))? > 0.

Furthermore, if there exists a solution x0, then it must be positive since

1(y) = h′

1(y) +

N(a − 1)2

ˆh1(0) = (a − 1)F−1(X0) < 0.

Theorem 4.1 will now follow by combining the following proposition with the arguments

explained in Section A.

Proposition B.3 The function C(1): Ξ → R has the strategy ξ(1)from Theorem 4.1 as its

unique minimizer. Moreover, the components of ξ(1)are all strictly positive.

21

hal-00166969, version 3 - 3 Feb 2010

Page 22

Proof: Thanks to Lemma B.1, there is at least one optimal strategy ξ∗= (x∗

standard results give the existence of a Lagrange multiplier ν ∈ R such that

∂

∂xiC(1)(x∗

0,...,x∗

N) ∈ Ξ, and

0,...,x∗

N) = ν for i = 0,...,N.

Now we use the form of C(1)as given in (47) to obtain the following relation between the partial

derivatives of C(1)for i = 0,...,N − 1:

∂

∂xiC(1)(x0,...,xN) = a

+ G′?aix0+ ··· + xi

Recalling (48), we obtain

?aix∗

Since h1is one-to-one we must have

?

∂

∂xi+1C(1)(x0,...,xN) − G′?a(aix0+ ··· + xi)??

?

h1

0+ ··· + x∗

i

?= ν (1 − a) for i = 0,...,N − 1.

x∗

x∗

x∗

N

0

= h−1

= x∗

= X0− x∗

1(ν (1 − a))

0(1 − a)

0− (N − 1)x∗

i

for i = 1,...,N − 1

0(1 − a).

(51)

Note that these equations link all the trades to the initial trade x0. Due to the dynamics (36)

and (37), it follows that the process E of ξ∗is given by

Etn= a(ax0+ x0(1 − a)) = ax0. (52)

Consequently, by (46),

C(1)(x∗

0,...,x∗

N)=G(x∗

+G?ax∗

0(x∗

0) − G(0) + (N − 1)?G(ax∗

N?G(x∗

=: C(1)

0+ x∗

0(1 − a)?− G(x∗

0(1 − a)) − G(ax∗

0)?

0+ X0− x∗

0) − G(x∗

0).

0− (N − 1)x∗

0a)

=

0a)?+ G?X0+ Nx∗

0(a − 1)?− G(0)

It thus remains to minimize the function C(1)

of an optimal strategy in Ξ for C(1), we know that C(1)

Differentiating with respect to y gives

0(y) with respect to y. Thanks to the existence

0(y) has at least one local minimum.

∂C(1)

0(y)

∂y

= N?F−1(y) − aF−1(ay) + (a − 1)F−1(X0+ Ny (a − 1))?

= Nˆh1(y).(53)

Lemma B.2 now implies that C(1)

exists. This local minimum must hence be equal to x∗

of the optimal strategy as well as our representation.

0

can only have one local minimum, which is also positive if it

0, which establishes both the uniqueness

22

hal-00166969, version 3 - 3 Feb 2010

Page 23

Finally, it remains to prove that all market orders in the optimal strategy are strictly

positive. Lemma B.2 gives ξ(1)

0

= x∗

As for the final market order, using the facts that (53) vanishes at y = x∗

increasing gives

0> 0 and then (51) gives ξ(1)

n = x∗

n> 0 for n = 1,...,N −1.

0and F−1is strictly

0 = F−1(x∗

> (1 − a)?F−1(ax∗

N> 0.

0) − aF−1(ax∗

0) − (1 − a)F−1(ax∗

0) − F−1(ax∗

0+ x∗

N)

0+ x∗

N)?,

which in turn implies x∗

C The optimal strategy in Model 2.

In this section, we will minimize the function C(2)of (43) over the set Ξ of all deterministic

strategies and thereby complete the proof of Theorem 5.1. To this end, recall first that the

definitions of D and E are given by (35)–(37). Based on their Model 2 dynamics, we will now

obtain a formula of the cost function C(2)of (43) in terms of the functions F,?F, and G, where

N

?

N

?

G is as in (45). For any deterministic strategy ξ = (x0,...,xN) ∈ Ξ,

C(2)(x0,...,xN) =

n=0

?Dtn+

?

Dtn

xf(x)dx

=

n=0

G(xn+ F (Dtn)) −?F (Dtn)

?

. (54)

We now state three technical lemmas that will allow to get the optimal strategy.

Lemma C.1 We have C(2)(x0,...,xN)−→+∞ for |ξ| → ∞, and therefore there exists a local

minimum of C(2)in Ξ.

Proof: We rearrange the sum in (54):

C(2)(x0,...,xN) =?F?aF−1(xN+ F(DtN))?

+

N

?

N

?

n=0

??F?F−1(xn+ F(Dtn))?−?F?aF−1(xn+ F(Dtn))??

??F?F−1(xn+ F(Dtn))?−?F?aF−1(xn+ F(Dtn))??

≥

n=0

.(55)

For the terms in (55), we have the lower bound

?F(z) −?F(az) =

????

?z

az

xf(x)dx

????≥1

23

2(1 − a2)z2

inf

? z∈[az,z]f(? z) ≥ 0.

hal-00166969, version 3 - 3 Feb 2010

Page 24

Let

H(y) =1

2(1 − a2)F−1(y)2

inf

x∈[aF−1(y),F−1(y)]f(x).

Then we have H(y)−→ + ∞ for |y| → ∞, due to (20) and (14). Besides, we have

C(2)(x0,...,xN) ≥ H(|T2(ξ)|∞)

where | · |∞denotes again the ℓ∞-norm on RN+1, and T2is the (nonlinear) transformation

T2(ξ) =?x0,x1+ F−1(Dt1),...,xN+ F−1(DtN)?.

It is sufficient to show that |T2(ξ)|∞−→∞ when |ξ| → ∞. To prove this, we suppose by

way of contradiction that there is a sequence ξksuch that |ξk|∞ −→ ∞ and T2(ξk) stays

bounded. Then, all coordinates in the sequence (T2(ξk))kare bounded, and in particular (xk

is a bounded sequence. Therefore, Dk

coordinate xk

manner, we get that (xk

contradiction.

0)k

t1= aF−1(xk

0) is also a bounded sequence. The second

1)kis a bounded sequence. In that

1+F−1(Dk

t1) being also bounded, we get that (xk

n)kis a bounded sequence for any n = 0,...,N, which is the desired

Lemma C.2 (Partial derivatives of C(2)).

We have the following recursive scheme for the derivatives of C(2)(x0,...,xN)

for i = 0,...,N − 1:

∂

∂xiC(2)= F−1(xi+ F(Dti)) +

af?Dti+1

?

f (F−1(xi+ F(Dti)))

?

∂

∂xi+1C(2)− Dti+1

?

. (56)

Proof: From (23) we get the following scheme for Dtnfor a fixed n ∈ {1,...,N}:

Dtn

?

aF−1(xn−1+F(Dtn−1))

?

...

aF−1(xi+1+F(Dti+1))

?

aF−1(xi+F(Dti))

?

...

aF−1(x0).

Therefore the following relation holds for the partial derivatives of Dtn:

∂

∂xiDtn=

af(Dti+1)

f (F−1(xi+ F(Dti)))

∂

∂xi+1Dtn,i = 0,...,n − 2.(57)

24

hal-00166969, version 3 - 3 Feb 2010

Page 25

Furthermore, according to (54) and (48),

∂

∂xiC(2)= F−1(xi+ F(Dti))+ (58)

+

N

?

n=i+1

f(Dtn)∂

∂xiDtn

?F−1(xn+ F(Dtn)) − Dtn

?

for i = 0,...,N. Combining (58) and (57) yields (56). Note that (57) is only valid up to

i = n − 2.

Lemma C.3 Under the assumptions of the Theorem 5.1, equation (21) has at most one solu-

tion x0> 0. Besides, the function g(x) := f(x) − af(ax) is positive.

Proof: Uniqueness will follow if we can show that both h2◦ F−1and

ˆh2(y) := −F−1?X0− N?y − F?aF−1(y)???

are strictly increasing. Moreover, h2◦F−1(0) = 0 andˆh2(0) < 0 so that any zero of h2◦F−1+ˆh2

must be strictly positive.

The function h2 is one-to-one, has zero as fixed point, and satisfies (26). It is therefore

strictly increasing, and since F−1is also strictly increasing, we get that h2◦ F−1is strictly

increasing. It remains to show thatˆh2is strictly increasing. We have that

ˆh′

2(y) = N

f (F−1(y)) − af (aF−1(y))

f (F−1(y))f (F−1(X0− N [y − F (aF−1(y))])),

is strictly positive, because, as we will show now, the numerator of this term is positive. The

numerator can be expressed as g(F−1(y)) for g as in the assertion. Hence, establishing strict

positivity of g will conclude the proof. To prove this we also define g2(x) := f(x) − a2f(ax) so

that

h2(x) = xg2(x)

g(x).

Both functions g and g2 are continuous and have the same sign for all x ∈ R due to the

properties of h2explained at the beginning of this proof. Because of g(x) < g2(x) for all x ∈ R,

we infer that there can be no change of signs, i.e., either g(x) > 0 and g2(x) > 0 for all x ∈ R

or g(x) < 0 and g2(x) < 0 everywhere. With g(0) = f(0)(1 − a) > 0 we obtain the positivity

of g.

Theorem 5.1 will now follow by combining the following proposition with the arguments

explained in Section A.

Proposition C.4 The function C(2): Ξ → R has the strategy ξ(2)from Theorem 5.1 as its

unique minimizer. Moreover, the components of ξ(2)are all strictly positive.

25

hal-00166969, version 3 - 3 Feb 2010

Page 26

Proof: The structure of the proof is similar to the one of Theorem 4.1 although the computations

are different. Thanks to Lemma C.1, we know that there exists an optimal strategy ξ∗=

(x∗

0,...,x∗

N) ∈ Ξ. There also exists a corresponding Lagrange multiplier ν such that

∂

∂xiC(2)(x∗

From (56), we get

?F−1(x∗

Since h2 is one-to one, this implies in particular that x∗

0,...,N − 1. It follows from (23) also Dti+= F−1(x∗

Dti+= Dt0+= F−1(x∗

0,...,x∗

N) = ν,i = 0,...,N.

ν = h2

i+ F (Dti))?,i = 0,...,N − 1.

i+ F (Dti) does not depend on i =

i+ F (Dti)) is constant in i, and so

Dti+1= aF−1(x∗

0) and

0). (59)

Hence,

x∗

x∗

x∗

N

0

= F?h−1

= X∗

2(ν)?,

0− x∗

i

= x∗

0− F(Dti) = x∗

0− F?aF−1(x∗

0)?

for i = 1,...,N − 1, (60)

0− (N − 1)?x∗

N) is equal to

0− F?aF−1(x∗

0)??.

These equations link all market orders to the initial trade x∗

we find that C(2)(x∗

0) := C(2)?

=NG(x∗

0. Using (60) and once again (59),

0,...,x∗

C(2)

0(x∗

x∗

0,x∗

0) −?F?aF−1(x∗

0− F(aF−1(x∗

0)),...,X0− Nx∗

0)??

0+ (N − 1)F(aF−1(x∗

0))

?

?

+ G?X0+ N?F?aF−1(x∗

0

and thus

?

0)?− x∗

0(x∗

??

0

??.

The initial trade x∗

0must clearly be a local minimum of C(2)

?

which is equivalent to

∂

∂yC(2)

0) = 0. Therefore,

0 = ND0+− a2D0+f(Dt1)

f(D0+)+ DtN+

af(Dt1)

f(D0+)− 1,

DtN+= D0+f(D0+) − a2f(Dt1)

f(D0+) − af(Dt1).(61)

This is just equation (21), which has at most one solution, due to Lemma C.3. This concludes

the proof of the existence and the representation of the optimal strategy ξ(2)in Theorem 5.1.

Finally, we need to show the strict positivity of the optimal strategy. Thanks to the posi-

tivity of the optimal x∗

0, we get

x∗

i= x∗

0− F(aF−1(x∗

0)) > 0

for i = 1,...,N −1. So it only remains to show that x∗

DtN+= D0+f(D0+) − a2f(aD0+)

f(D0+) − af(aD0+)= D0+

The fraction on the right is strictly positive due to Lemma C.3. Hence,

DtN+> D0+=1

N> 0. We infer from (61) and (59) that

1 +af(aD0+) − a2f(aD0+)

f(D0+) − af(aD0+)

?

?

.

aDtN> DtN,

which implies x∗

N> 0.

26

hal-00166969, version 3 - 3 Feb 2010

Page 27

D Optimal strategy for block-shaped LOB.

Here we prove the results of Section 6.

Our aim is to prove Proposition 6.2, i.e., to show that the strategy (27) satisfies the re-

cursion (31). The key point is that we have indeed explicit formulas for the coefficients in the

backward schemes of Proposition 6.2.

Lemma D.1 The coefficients αn, βn, and γnfrom (33) are explicitly given by

αn =

(1 + a−1) − qλ[(N − n)(a−1− 1) + 2(1 + a−1)]

2q[(N − n)(a−1− 1) + (1 + a−1)]

1 + a−1

[(N − n)(a−1− 1) + (1 + a−1)]

(N − n)(1 − a−1)

2κ[(N − n)(a−1− 1) + (1 + a−1)].

(62)

βn =

γn =

The explicit form of the sequences δn, ǫnand φnfrom (32) is

δn =

2a−2[(N − n)(a−1− 1) + (1 + a−1)]

κ[(N − n)(1 − a−2) + (N − n + 2)(a−3− a−1)]

κ(a−1− a)

[(N − n)(a−1− 1) + (1 + a−1)]

(N − n + 1)(a−1− a) − (N − n)(1 − a2)

[(N − n)(a−1− 1) + (1 + a−1)]

(63)

ǫn =

φn =

.

This Lemma can be proved in two steps. First, by a backward induction, we get the explicit

formulas for α, β and γ. Then, combining (62) with (33) and (32), we get (63).

Proof of Proposition 6.2. We can deduce the following formulas from the preceding lemma:

δnǫn=

2

(N − n)(1 − a) + 2,δnφn=2

κ

(N − n)(1 − a) + 1

(N − n)(1 − a) + 2. (64)

They will turn out to be convenient in (31).

Let us now consider the optimal strategy (ξ∗

processes Dt:= DA

0,...,ξ∗

N) from (27). We consider the associated

tand Xtas defined in (28) and (30). For n = 0, we have

ξ∗

0=

X0

(N − 1)(1 − a) + 2=12δ1ǫ1

and it satisfies (31) because D0= 0. For n ≥ 1, we can show easily by induction on n that

Dtn= aκξ∗

0. From (27), we get that ξ∗

n= (1 − a)ξ∗

0for n ?∈ {0,N}, and therefore we get

0− (n − 1)(1 − a)ξ∗

Xtn= X0− ξ∗

0= [(N − n)(1 − a) + 1]ξ∗

0.

Using these formulas, and combining with (64), it is now easy to check that

for n ∈ {1,...,N − 1},

ξ∗

2[δn+1ǫn+1Xtn− δn+1φn+1Dtn],

which shows that the optimal strategy given in (27) solves (31).

n=1

27

hal-00166969, version 3 - 3 Feb 2010

Page 28

Acknowledgement. Support from the Deutsche Bank Quantitative Products Laboratory

is gratefully acknowledged. The authors thank the Quantitative Products Group of Deutsche

Bank, in particular Marcus Overhaus, Hans B¨ uhler, Andy Ferraris, Alexander Gerko, and Chrif

Youssfi for stimulating discussions and useful comments (the statements in this paper, however,

express the private opinion of the authors and do not necessarily reflect the views of Deutsche

Bank). Moreover, it is a pleasure to thank Anna Obizhaeva and Torsten Sch¨ oneborn for helpful

comments on earlier versions of this paper.

References

[1] Alfonsi, A., Fruth, A., Schied, A. Constrained portfolio liquidation in a limit order book

model. Banach Center Publ. 83, 9-25 (2008).

[2] Almgren, R. Optimal execution with nonlinear impact functions and trading-enhanced

risk, Applied Mathematical Finance 10 , 1-18 (2003).

[3] Almgren, R., Chriss, N. Value under liquidation. Risk, Dec. 1999.

[4] Almgren, R., Chriss, N. Optimal execution of portfolio transactions. J. Risk 3, 5-39

(2000).

[5] Almgren, R., Lorenz, J. Adaptive arrival price. In: Algorithmic Trading III: Precision,

Control, Execution, Brian R. Bruce, editor, Institutional Investor Journals (2007).

[6] Almgren, R., Thum, C. Hauptmann, E., Li, E. Equity market impact. Risk, July (2005).

[7] Bank, P., Baum, D. Hedging and portfolio optimization in financial markets with a large

trader. Math. Finance 14, no. 1, 1–18 (2004).

[8] Bertsimas, D., Lo, A. Optimal control of execution costs. Journal of Financial Markets,

1, 1-50 (1998).

[9] Biais, B., Hillion, P., Spatt, C. An empirical analysis of the limit order book and order

flow in Paris Bourse. Journal of Finance 50, 1655-1689 (1995).

[10] Bouchaud, J. P., Gefen, Y., Potters, M. , Wyart, M. Fluctuations and response in finan-

cial markets: the subtle nature of ‘random’ price changes. Quantitative Finance 4, 176

(2004).

[11] Brunnermeier, M., Pedersen, L. Predatory trading. Journal of Finance 60, 1825-1863

(2005).

[12] Carlin, B., Lobo, M., Viswanathan, S. Episodic liquidity crises: Cooperative and preda-

tory trading. Forthcoming in Journal of Finance.

[13] Cetin, U., Jarrow, R., Protter, P. Liquidity risk and arbitrage pricing theory. Finance

Stoch. 8 , no. 3, 311–341 (2004).

28

hal-00166969, version 3 - 3 Feb 2010

Page 29

[14] Frey, R. Derivative asset analysis in models with level-dependent and stochastic volatility.

Mathematics of finance, Part II. CWI Quarterly 10 , no. 1, 1–34 (1997).

[15] Frey, R., Patie, P. Risk management for derivatives in illiquid markets: a simulation

study. Advances in finance and stochastics, 137–159, Springer, Berlin, 2002.

[16] Obizhaeva, A., Wang, J. Optimal Trading Strategy and Supply/Demand Dynamics,

Preprint, forthcoming in Journal of Financial Markets.

[17] Potters, M., Bouchaud, J.-P. More statistical properties of order books and price impact.

Physica A 324, No. 1-2, 133-140 (2003).

[18] Schied, A., Sch¨ oneborn, T. Optimal basket liquidation with finite time horizon for CARA

investors. Preprint, TU Berlin (2008).

[19] Schied, A., Sch¨ oneborn, T. Risk aversion and the dynamics of optimal liquidation strate-

gies in illiquid markets. To appear in Finance and Stochastics.

[20] Sch¨ oneborn, T., Schied, A. Competing players in illiquid markets: predatory trading vs.

liquidity provision. Preprint, TU Berlin.

[21] Weber, P., Rosenow, B. Order book approach to price impact. Quantitative Finance 5,

no. 4, 357-364 (2005).

29

hal-00166969, version 3 - 3 Feb 2010