Optimal trade execution and absence of price manipulations in limit order book models

Abstract

We analyze the existence of price manipulation and optimal trade execution strategies in a model for an electronic limit order book with nonlinear price impact and exponential resilience. Our main results show that, under general conditions on the shape function of the limit order book, placing deterministic trade sizes at trading dates that are homogeneously spaced is optimal within a large class of adaptive strategies with arbitrary trading dates. This extends results from our earlier work with A. Fruth. Perhaps even more importantly, our analysis yields as a corollary that our model does not admit price manipulation strategies. This latter result contrasts the recent findings of Gatheral [12], where, in a related but different model, exponential resilience was found to give rise to price manipulation strategies when price impact is nonlinear.

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Optimal execution and absence of price manipulations in
limit order book models
Aur´ elien Alfonsi
Universit´ e Paris-Est, CERMICS,
Project team MathFi ENPC-INRIA-UMLV,
Ecole des Ponts, 6-8 avenue Blaise Pascal,
77455 Marne La Vall´ ee, France
alfonsi@cermics.enpc.fr
Alexander Schied
Zentrum Mathematik
TU Munich
Boltzmannstr. 3
85748 Munich, Germany
schied@tum.de
June 22, 2009
Abstract: We continue the analysis of optimal execution strategies in the model for a limit
order book with nonlinear price impact and exponential resilience that was considered in our
earlier paper with A. Fruth [2]. We now allow for non-homogeneous resilience rates and arbitrary
trading dates and consider the extended problem of optimizing jointly over trading dates and
sizes. Our main results show that, under general conditions on the shape function of the
limit order book, placing the deterministic trade sizes identified in [2] at trading dates that are
homogeneously spaced is optimal also within the large class of adaptive strategies with arbitrary
trading dates. Perhaps even more importantly, our analysis yields as a corollary that our model
does not admit price manipulation strategies in the sense of Huberman and Stanzl [14]. This
latter result contrasts the recent findings of Gatheral [13], where, in a related but different
model, exponential resilience was found to give rise to price manipulation strategies when price
impact is nonlinear.
1Introduction.
The problem of optimal execution is concerned with the optimal acquisition or liquidation of
large asset positions. In doing so, it is usually beneficial to split up the large order into a
sequence of partial order, which are then spread over a certain time horizon, so as to reduce
the overall price impact and the execution costs. The optimization problem at hand is thus
to find a trading strategy that minimizes a cost criterion under the constraint of overall order
execution within a given time frame. There are several reasons why studying this problem is
interesting.
First, liquidity risk is one of the least understood sources of financial risk, and one of its
various aspects is the risk resulting from price impact created by trading large positions. Due
to the nonlinear feedback effects on dynamic trading strategies, market impact risk is probably
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also among the most fascinating aspects of liquidity risk for mathematicians. The optimal
execution problem allows studying market impact risk in its purest form. Moreover, the results
obtained for this problem can serve as building blocks in a realistic analysis of more complex
problems such as the hedging of derivatives in illiquid markets.
Second, the mathematical analysis of optimal execution strategies can help in the ongo-
ing search for viable market impact models. As argued by Huberman and Stanzl [14] and
Gatheral [13], any reasonable market impact model should not admit price manipulation strate-
gies in the sense that there are no ‘round trips’ (i.e., trading strategies with zero balance in
shares), whose expected trading costs are negative. Since every round trip can be regarded as
the execution of a zero-size order, a solution of the optimal execution problem also includes an
analysis of price manipulation strategies in the model (at least as limiting case when the order
size tends to zero).
In recent years, the problem of optimal execution was considered for various market impact
models and cost functions by authors such as Bertsimas and Lo [10], Almgren and Chriss
[5, 6], Almgren [4], Obizhaeva and Wang [16], Almgren and Lorenz [7], the second author and
Sch¨ oneborn [18, 19], our joint papers with A. Fruth [1, 2], and Bayraktar and Ludkovski [9], to
mention only a few.
Here, we continue our analysis from [1, 2]. The underlying model is a time-inhomogeneous
version of the one in [2] and thus an extension of the limit order book model with linear impact
and exponential resilience that was proposed by Obizhaeva and Wang [16]. In our model, the
ask part of the limit order book consists of a certain distribution of shares offered in the form
of limit orders at prices higher than the current best ask price. When the large trader has not
yet been active, the best ask price fluctuates according to the actions of noise traders. A buy
market order placed by the large trader consumes a block of shares located immediately to the
right of the best ask and thus increases the ask price by a certain amount, the price impact.
Since the distribution of limit orders is allowed to be non-uniform, the price impact created by
a market order is typically a nonlinear function of the order size. In reaction to the increased
bid-ask spread, the best ask price will recover from the impact of the buy order within a certain
time span, i.e., it will show a certain resilience. Thus, the price impact of a market order will
neither be completely instantaneous nor entirely permanent but will decay exponentially with
a time-dependent resilience rate. As in [2], we consider the following two distinct possibilities
for modeling the resilience of the limit order book after a large market order: the exponential
recovery of the number of limit orders, i.e., of the volume of the limit order book (Model 1), or
the exponential recovery of the bid-ask spread (Model 2).
This model is quite close to descriptions of price impact on limit order books found in
empirical studies such as Biais et al. [11], Potters and Bouchaud [17], Bouchaud et al. [12], and
Weber and Rosenow [20]. 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. As
for the order of the immediate price impact, our assumptions include in particular the cases of
power-law impact as proposed in Almgren et al. [8] and of logarithmic impact as suggested in
Bouchaud et al. [12].
While in [2] we considered only strategies whose trades are placed at equidistant times,
we now allow the trading times to be stopping times. This problem description is clearly
much more natural than prescribing a priori the dates at which trading may take place. It is
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also more realistic than the idealization of trading in continuous time. In addition, the time-
inhomogeneous description allows us to account for time-varying liquidity and thus in particular
for the well-known U-shape patterns in intraday market parameters; see, e.g., [15].
Optimal execution in this extended framework leads to the problem of optimizing simultane-
ously over both trading times and sizes. This problem is more complex than the one considered
in [2] and requires new arguments. Nevertheless, our main results show that the unique opti-
mum is attained by placing the deterministic trade sizes identified in [2] at trading dates that
are homogeneously spaced with respect to the average resilience rate in between trades.
As a corollary, we show that neither of the two variants of our model admits price manipu-
lation strategies in the sense of Huberman and Stanzl [14] and Gatheral [13], provided that the
shape function of the limit order book belongs to a certain class of functions (which slightly
differs for each variant). In either case, this class of functions contains the power-law family
and logarithmic price impact. This corollary is fairly surprising and puzzling in view of recent
results by Gatheral [13]. There it was shown that, in a closely related market impact model,
exponential resilience leads to the existence of price manipulation strategies as soon as price
impact is nonlinear.
As of now, it is not clear to us which of the features in Gatheral’s model that are different
from ours are responsible for the creation of price manipulation strategies. In fact, one of our
motivations for the research reported in this paper was to understand whether the restriction
to equidistant time grids in [2] prevented the emergence of price manipulation strategies in our
model.
In view of our results on the nonexistence of price manipulation strategies, we must therefore
deduce that, in contrast to Gatheral’s conclusion, exponential resilience of a limit order book
is a viable possibility for describing the decay of market impact, at least from a theoretical
perspective.
This paper is organized as follows. In Section 2.1 we introduce the limit order book model
with its two variants. The cost optimization problem is explained in Section 2.2. In Section 2.3
we state our main results for the case of a block-shaped limit order book, which corresponds
to linear price impact. This special case is much simpler than the case with nonlinear price
impact. We therefore give a self-contained description and proof for this case, so that the reader
can gain a quick intuition on why our results are true. The proofs for the block-shaped case
rely on the results from our earlier paper [1] with A. Fruth and are provided in Section 3.2.
As in [2], we refer to the variant of our model with exponential resilience for the volume of
the limit order book as Model 1. The main results for this case are stated in Section 2.4. The
corresponding proofs are given in Section 3.3. Model 2 refers to the exponential resilience of
the bid-ask spread. The results for this variant are stated in Section 2.5, while proofs are given
in Section 3.4.
2 Setup and main results
In this section we first introduce the two variants of our model and state the optimization
problem. We then state our results for the particularly simple case of a block-shaped limit
order book. Subsequently, we formulate our theorems for each model variant individually.
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2.1Model description
The model variants that we consider here are time-inhomogeneous versions of the ones in [2].
The general aim is to model the dynamics of a limit order book that is exposed to repeated
market orders by a large trader, whose goal is 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 limit order book, which consists of shares offered at various ask
prices. The lowest ask price at which shares are offered is called the best ask price. Symmetric
statements hold of course for the liquidation of a given asset position of X0> 0 shares.
When the large trader is inactive, 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 an arbitrary martingale on a given filtered probability space (Ω,(Ft),F,P) and satisfies
A0
0= A0.
Above the unaffected best ask price A0
available shares in the limit order book: the number of shares offered at price A0
by f(x)dx for a bounded and continuous density function f : R\{0} −→ (0,∞). We call f the
shape function of the limit order book. It determines the impact of a market order placed by
our large trader. The choice of a constant shape function corresponds to linear market impact
and to the block-shaped limit order book model of Obizhaeva and Wang [16].
The actual best ask price at time t, i.e., the best ask price after taking the price impact
of previous buy orders of the large trader into account, is denoted by At. It is clearly above
the unaffected best ask price, and the extra spread caused by the actions of the large trader is
denoted by
DA
t, we assume a continuous ask price distribution for
t+ x is given
t:= At− A0
t.
A buy market order of xt> 0 shares placed by the large trader at time t will 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.
Thus, the process DAcaptures the impact of market orders on the current best ask price.
Clearly, the price impact DA
constant between DA
t+−DA
t+.
twill be a nonlinear function of the order size xtunless f is
tand DA
Example 2.1. Consider the shape functions of the form
f(x) =









q+
(1 + λ+x)α+
for x > 0,
q−
(1 − λ−x)α−
for x < 0,
where q±and λ±are positive constants and α±∈ (0,1]. For α+= 1, DA
rithmically on the trade size xt, a functional dependence suggested by Bouchaud et al. [12].
t+will depend loga-
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For 0 < α+ < 1 we get the power law impact found by Almgren et al. [8]. The particular
case α+= 1/2 corresponds to the popular square-root dependence; see [13, Section 6.2] for a
discussion. The parameters q−, λ−, and α−will of course determine the shape of the limit order
book for sell orders.
The quantity
EA
t=
?DA
t
0
f(x)dx, (1)
describes the volume impact in the limit order book present at time t. By introducing the
antiderivative
F(z) =
0
of f, the relation (1) can also be expressed as
?z
f(x)dx(2)
EA
t= F(DA
t)andDA
t= F−1(EA
t),
where we have used our assumption that f is strictly positive to obtain the second identity.
For simplicity, the function F is supposed to be unbounded in the sense that
lim
x↑∞F(x) = ∞ and lim
x↓−∞F(x) = −∞,(3)
i.e., we assume that the limit order book has infinite depth. Relaxing this assumption is possible
but would require constraints on the order sizes and thus complicate the problem description.
When the large trader is inactive in between market orders, DAand EArevert back to
zero on an exponential scale with a deterministic, time-dependent rate t ?→ ρt, called resilience
speed. More precisely, we will consider the following two model variants for the resilience of the
market impact:
Model 1: EAevolves according to
dEA
t= −ρtEA
tdt
if the large investor is not placing buy orders during the time interval [t,t + s).
Model 2: DAevolves according to
dDA
t= −ρtDA
tdt
if the large investor is not placing buy orders during the time interval [t,t + s).
With these features we model the well-established empirical fact that order books exhibit a
certain resilience as to the price impact of a large buy market orders. That is, after the initial
impact the best ask price reverts back to its previous position; cf. Biais et al. [11], Potters and
Bouchaud [17], Bouchaud et al. [12], and Weber and Rosenow [20] for empirical studies.
Up to now, we have only described the effect of buy orders on the upper half of the limit
order book. 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 it could be beneficial to also sell some shares and to buy them back at
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