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Optimal starting times, stopping times and risk measures for
algorithmic trading: Target Close and Implementation Shortfall
arXiv:1205.3482v6 [q-fin.TR] 16 Dec 2013
Mauricio Labadie
1 3
Charles-Albert Lehalle
2 3
December 17, 2013
Abstract
We derive explicit recursive formulas for Target Close (TC) and Implementation Shortfall (IS) in
the Almgren-Chriss framework. We explain how to compute the optimal starting and stopping times
for IS and TC, respectively, given a minimum trading size. We also show how to add a minimum
participation rate constraint (Percentage of Volume, PVol) for both TC and IS.
We also study an alternative set of risk measures for the optimisation of algorithmic trading curves.
We assume a self-similar process (e.g. L´vy process, fractional Brownian motion or fractal process)
e
and define a new risk measure, the
p-variation,
which reduces to the variance if the process is a Brown-
ian motion. We deduce the explicit formula for the TC and IS algorithms under a self-similar process.
We show that there is an two-way relationship between self-similar models and a family of risk
measures called
p-variations.
indeed, it is equivalent to have (1) a self-similar process and calibrate
empirically the parameter
p
for the
p-variation,
or (2) a Brownian motion and use the
p-variation
as
risk measure instead of the variance. We also show that
p
can be seen as a fine-tuning parameter
which modulates the aggressiveness of the trading protocole:
p
increases if and only if the TC algo-
rithm starts later and executes faster.
Finally, we show how the parameter
p
of the
p-variation
can be implied from the optimal starting
time of TC. Under this framework
p
can be viewed as a measure of the joint impact of market impact
(i.e. liquidity) and volatility.
Keywords:
Quantitative Finance, High-Frequency Trading, Algorithmic Trading, Optimal Execution,
Market Impact, Risk Measures, Self-similar Processes, Fractal Processes.
Corresponding author. EXQIM (EXclusive Quantitative Investment Management). 24 Rue de Caumartin 75009 Paris,
France. Email: mauricio.labadie@gmail.com.
2
CFM (Capital Fund Management). 23 Rue de l’Universit´ 75007 Paris, France.
e
3
This research was mostly done when both authors were working at Cr´dit Agricole - Cheuvreux (now Kepler
e
Cheuvreux).
1
1
2
M. Labadie
·
C.A. Lehalle
Contents
1 Introduction
2 Optimal starting and stopping times
2.1 A review of the mean-variance optimisation of Almgren-Chriss
2.2 The Shooting Method . . . . . . . . . . . . . . . . . . . . . . .
2.3 Derivation of the Target Close (TC) algorithm . . . . . . . . .
2.4 Derivation of the Implementation Shortfall (IS) algorithm . . .
2.5 Comparison between TC and IS . . . . . . . . . . . . . . . . . .
2.6 Adding constraints: Percentage of Volume (PVol) . . . . . . . .
2.7 Computing the optimal stopping time for TC . . . . . . . . . .
2.8 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Non-Brownian models: self-similar processes
3.1 The
p-variation
model . . . . . . . . . . . . . . . .
3.2 Optimal trading algorithms using
p-variance
as risk
3.3 Examples of self-similar processes . . . . . . . . . .
3.4 Numerical results . . . . . . . . . . . . . . . . . . .
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4 Assessing the effects of the risk measure
4.1 Equivalence between risk measures and models . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Risk measures, starting times and slopes for the TC algorithm . . . . . . . . . . . . . . .
4.3 Implied
p-variation
for CAC40 and link with liquidity . . . . . . . . . . . . . . . . . . . .
5 Final remarks
Optimal starting times, stopping times and risk measures for algorithmic trading
3
1
Introduction
Purpose of the paper.
Trading algorithms are used by asset managers to control the trading rate of
a large order, performing a balance between trading fast to minimise exposure to market risk and trading
slow to minimise market impact (for an overview of quantitative trading methods, see [Lehalle, 2012]
and [Abergel et al., 2012]). This balance is usually captured via a cost function which takes into account
two joint effects, namely the market impact and the market risk. The first frameworks to be proposed
were [Bertsimas and Lo, 1998] and [Almgren and Chriss, 2000], the latter using a mean-variance crite-
ria. More sophisticated cost functions have been already proposed in the academic literature, leading
to the use of different optimization approaches like stochastic control (see [Bouchard et al., 2011] or
[Gu´ant et al., 2011]) or stochastic algorithms (see [Pag`s et al., 2012]).
e
e
From the practitioners’ viewpoint, the cost function to choose is far from obvious. The easiest way
to proceed is to replace the choice of the cost function by observable features of the market. This is the
approach chosen in this paper, where the cost function generalises the mean-variance frameworks of both
Almgren-Chriss and [Gatheral et al., 2010], which a mean-p-variation. Instead of complex cost functions
and compute-intensive parameter calibration, this paper proposes a simpler approach that covers a large
class of parametrized cost function already in use by practitioners. We calibrate the parameters from
observable variables like stopping times and maximum participation rates.
This approach is very flexible and customisable. Indeed, since it depends on a single fine-tuning
parameter
p,
a practitioner can either calibrate
p
or modify it by hand to fit their risk budget. A
good example is the maximum participation rate: actually most practitioners are using a mean-variance
criteria with an arbitrary risk aversion parameter, but add a “control layer” to their algorithms in order
to ensure that the participation on real time will never be more than a pre-determined threshold (i.e.
the trading algorithm will never buy or sell more than a certain percentage of the volume traded by the
whole market); here we propose a way to include this constraint into the full optimisation process, at the
very first step of the process. Moreover, some traders know that they would like to see a given algorithm
finish a buy of a given number of shares within a certain time period; again; we propose a way to implicit
the parameters of the cost function to achieve this.
An optimal trading framework for the target close and implementation shortfall benchmarks
with percentage of volume constraints.
A TC (Target Close) algorithm is a trading strategy that aims to execute a certain amount of shares as
near as possible to the closing auction price. Since the benchmark with respect to which the TC algorithm
is measured is the closing price, the trader has interest in executing most of their order at the close auction.
However, if the number of shares to trade is too large, the order cannot be totally executed at the
close auction without moving the price too much due to its market impact [Gatheral and Schied, 2012].
Therefore, the trader has to trade some shares during the continuous auction phase (i.e. before the close)
following one of the now well-known optimal trading algorithms available, e.g. mean-variance optimisation
(following [Almgren and Chriss, 2000]) or stochastic control (like in [Bouchard et al., 2011]).
As we have mentioned above, this paper will stay close to the original Almgren-Chriss framework,
extending the risk measure from the variance to a general
p-variation
criterion. The goal of this paper
to explain the practical interpretation of the
p-variation
parameter used in the optimisation scheme and
show how to choose them optimally in practice.
The
p-variation
is an extension of the variance, depending on
p,
since when
p
= 2 we recover the
variance. When
p
= 2 the trader assumes that (1) the price is no longer a martingale, i.e. there are
patterns in prices (trend-following or mean-reverting), and (2) the time-scaling properties of prices are
4
M. Labadie
·
C.A. Lehalle
not as in the Brownian motion. Therefore, a new risk measure other than variance is needed. This paper
explores the impact of
p
on the properties of the obtained optimal trading curve, and relates it with
self-similar processes (e.g. fractional Brownian motion, L´vy processes and multifractal processes).
e
Inverting the optimal liquidation problem putting the emphasis on observables of the ob-
tained trading process.
We will show that the TC (Target Close) algorithm can be seen as a “reverse
IS
” (Implementation
Shortfall) –see equation (12) and following for details–. In this framework, the starting time for a TC
is as important than the ending time for an IS. For practitioners this distinction is even more critical
since shortening the trading duration of an IS because of an interesting price opportunity can always be
justified, but beginning sooner or later than an “expected
optimal start time”
for a TC is more difficult
to explain.
The paper also shows that the results obtained for the TC criterion can be applied to the IS criterion
because TC and IS are both sides of the same coin. Indeed, on the one hand, the TC has a pre-determined
end time, its benchmark is the price at the end of the execution and the starting time is unknown. On
the other hand, IS has a pre-determined starting time, its benchmark is the price at the beginning of the
execution and the stopping time is unknown. Therefore, there is no surprise that the recursive formula
for IS turns out to be exactly the same that for TC but with the time running backwards.
It is customary for practitioners to put constraints on the maximum participation rate of a trading
algorithms (say 20% of the volume traded by the market). Therefore, it is of paramount importance
to find a systematic way of computing the starting time of a TC under a percentage of volume (PVol)
constraint. Such an “optimal
trading policy under PVol constraint”
is properly defined and solved in
this paper. A numerical example with real data is provided, where the optimal trading curves and their
corresponding optimal starting times are computed.
Solving the TC problem under constraints allows us to analyze the impact of the parameters of the
optimisation criterion on observable variables of the trading process. It should be straightforward for
quantitative traders the task to implement our results numerically, i.e. to choose the characteristics of
the trading process they would like to target and then infer the proper value of the parameters of the
criterion they need.
Link between a mean
p-variation
criterion and self similar price formation processes.
[Almgren and Chriss, 2000] developed a mean-variance framework to trade IS (Implementation Short-
fall) portfolios driven by a Brownian motion. More recently, [Lehalle, 2009] extended the model to
Gaussian portfolios whilst [Gatheral and Schied, 2012] addressed the same problem for the geometric
Brownian motion. In this article we extend the analysis to a broad class of non-Brownian models, the
so-called self-similar models, which include L´vy processes and fractional Brownian motion (for empiri-
e
cal studies about the self-similarity of intraday data, see [Xu and Gen¸ay, 2003], [M¨ller et al., 1990] or
c
u
[Cont et al., 1997]). We study in detail the relationship between the exponent of self-similarity, the choice
of the risk measure and the level of aggressiveness of the algorithm. We show that there are two opposite
approaches that nevertheless give the same recursive trading formula: one assumes a self-similar process,
estimates the exponent of self-similarity
H
and chooses the
p-variation
via
p
= 1/H; the other assumes
a classical Brownian motion and chooses the
p-variation
as the risk measure instead of the the variance.
In the same way the starting time of a TC or the ending time of an IS can be used as an observable to
infer values of parameters of the optimization program, the maximum participation rate is expressed as
a function of
p
for a mean
p-variance
criterion. In the light of this, a quantitative trader who has chosen
Optimal starting times, stopping times and risk measures for algorithmic trading
5
to trade no more than 30% of the market volume during a given time interval, can modifie the value of
p
to fine-tune their execution and respect their constraints.
This paper formalizes an innovative approach of optimal trading based on observable variables, risk
budget and participation constraints. By doing so, it opens the door to a framework close to risk neutral
valuation of derivative products in optimal trading: instead of choosing the measure under which to
compute the expectation of the payoff (because optimal trading is always considered under historical
measure), we propose to infer the value of some parameters of the cost function so that the trading
process will satisfy some observable characteristics (start time, end time, maximum participation rate,
etc). In this framework, instead of being hedged with respect to market prices, the trader will be hedged
with respect to the risk-performance profile of an ideal trading process, i.e. a proxy she defined a priori.
Notice that we have chosen to extend the usual mean-variance criterion rather than going to more non-
parametric approaches like stochastic control. The main reason for this approach is because our framework
allows more explicit recurrent formulas, not to mention that our method can be easily extended to other
execution algorithms besides TC and IS.
Organisation of the article.
In Section 2 we derive a nonlinear, explicit recursive formula for both the TC and IS algorithms with a
nonlinear market impact. We explain how to build a TC algorithm under a maximum participation rate
constraint (percentage of volume, PVol). We provide a numerical example using real data, in which we
computed the trading curves and their optimal starting time. All our computations can be also applied
to IS.
In Section 3 we extend the analysis for a class of non-Brownian models called self-similar processes,
which include L´vy Processes, fractional Brownian motion and fractal processes. We define an
ad hoc
risk
e
measure, denoted
p-variation,
which renders the cost functional linear in time. We show numerically that
the exponent of self-similarity
H
can be viewed as a fine-tuning parameter for the level of aggressiveness
of the TC algorithm under PVol constraint.
In Section 4 we assess the effect of the parameter
p
in terms of risk management. We show the
existence of an equivalence between risk measures of
p-variation
type and self-similar models of exponent
H:
choosing a self-similar model, estimating
H
and defining
p
= 1/H for the risk measure yields the same
trading curve as assuming a Brownian motion but changing the risk measure from variance to
p-variation.
We also study the effect of
p
on the starting times for TC and the slopes of the corresponding trading
curves
We conclude by showing how the parameter
p
of the
p-variation
can be implied from the optimal
starting time of TC. In that framework
p
can be viewed measure of the joint impact of market impact
(i.e. liquidity) and volatility.
2
2.1
Optimal starting and stopping times
A review of the mean-variance optimisation of Almgren-Chriss
This section recalls the framework, notation and results in [Almgren and Chriss, 2000] and [Lehalle, 2009].
Suppose we want to trade an asset
S
throughout a time horizon
T >
0. Assume that we have already
set the trading schedule, i.e. we will do
N
trades at evenly distributed times
0 =
t
0
< t
1
< t
2
<
· · ·
< t
N
=
T.

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