Optimal execution strategies in limit order books with general shape functions
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, 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.
- SourceAvailable from: Erhan Bayraktar[Show abstract] [Hide abstract]
ABSTRACT: We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power-law and exponential-decay order book models and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.Mathematical Finance 01/2014; · 1.25 Impact Factor
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ABSTRACT: In limit order book markets, traders face the problem whether to display or hide their orders. While hiding reduces exposure impact, exposing can increase execution priority. Based on order flow dynamics, we develop a structural model that captures this trade-off. A central aspect of this work is the market impact of exposure: exposed limit orders affect incoming order flow. We derive explicit characterizations of the optimal exposure strategy under various market specifications. Using high-resolution ITCH data, we estimate the true impact of the exposure and calculate the optimal exposure for various stocks under high-frequency trading horizons. It turns out that exposure does mainly affect the supply side of liquidity. Our results suggests that the use of hidden orders can significantly increase trade performance.05/2014;
Article: Trading with Small Price Impact[Show abstract] [Hide abstract]
ABSTRACT: An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth. In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market, cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to optimal execution problems and to other market frictions such as proportional and fixed transaction costs.02/2014;
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
Quantitative Products Laboratory
10178 Berlin, Germany
Department of Mathematics, MA 7-4
Strasse des 17. Juni 136
10623 Berlin, Germany
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).
hal-00166969, version 3 - 3 Feb 2010
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 , Almgren and Chriss [3, 4],
Almgren and Lorenz , Obizhaeva and Wang , and Schied and Sch¨ oneborn [18, 19] to
mention only a few. For extensions to situations with several competing traders, see , ,
, 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 , , , , , , and . Also the
market impact models described in Bank and Baum , Cetin et al. , Frey , and Frey
and Patie  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  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  is quite close to descriptions of price impact on LOBs found in empirical
studies such as Biais et al. , Potters and Bouchaud , Bouchaud et al. , and Weber
and Rosenow . 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  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. ; see also Almgren  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 , 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 .
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
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 , 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 . 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
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 . In particular, we provide our new
explicit formula for the optimal strategy in a block-shaped LOB as obtained in . 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 .
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
Consequently, the ask price will be shifted up from A0to
t= A0+ σWtfor an (Ft)-Brownian motion W, as considered in .
t, we assume a continuous ask price distribution for
t+ x is given by f(x)dx
0+is determined by
f(x)dx = x0.
A0+:= A0+ DA
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
t:= At− A0
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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+is determined by the condition
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  and Almgren et al.  for a discussion.
Another important quantity is the process
t will be a nonlinear function of the order size xt unless
t+. Hence, our model includes the case of nonlinear impact
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
of f, the relation (2) can also be expressed as
where we have used our assumption that f is strictly positive to obtain the second identity.
The relation (1) is equivalent to
t+ xt. (5)
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