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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).

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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

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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

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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

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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)

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