Dynamic Optimal Execution Models with Transient Market ...
Transcript of Dynamic Optimal Execution Models with Transient Market ...
Original Paper
Dynamic Optimal Execution Models with Transient
Market Impact and Downside Risk
Yuhei Ono †1 , Norio Hibiki †2 , Yoshiki Sakurai †3
Abstract: When institutional investors trade large amounts of stock in the market, the trading
amount might impact the price, and this price change is called market impact (MI hereafter). In
addition, their trading is always exposed to uncertain price change, and this is called timing risk.
Such investors need to quantitatively evaluate the MI and timing risk, and decide the optimal execu-
tion strategy when considering the trade-off between them. Several previous studies assume tempo-
rary/permanent MI, while some recent studies discuss transient MI. On the contrary, most investors
need to manage their downside risk when executing an order to meet the trading needs below a target
cost. In this study, we discuss dynamic optimization models with transient MI and downside risk in
order to decide the optimal execution strategy. Specifically, we propose the following three types of
models based on Takenobu and Hibiki (2016) who assume temporary/permanent MI.
(1) Multiperiod model with step function using Monte Carlo simulation (Step model);
(2) Multiperiod model with piecewise linear (PwL) function based on the Step model; and
(3) One-period iterative model with static execution strategy (Iterative model).
We solve the optimal execution problems using these models, and conduct a sensitivity analysis to
examine the benefits of the models. In addition, we compare the three models, and evaluate their
characteristics and differences. We estimate the MI function and other parameters using market data,
and derive the optimal execution strategies for practical use.
Key words: dynamic optimal execution, transient market impact, market order, downside
risk
1 INTRODUCTION
When institutional investors trade large amounts
of stock in the market, the trading amount might
impact the price, and this price change is called
market impact (MI hereafter). In addition, their
trading is always exposed to uncertain price change,
and this is called timing risk. Such investors need
to quantitatively evaluate the MI and timing risk,
and decide the optimal execution strategy when
considering the trade-off between them. Several
previous studies assume temporary/permanent MI.
Bertsimas and Lo [1] derived the optimal strategy
†1This research is done in the Graduate School of Science
and Technology, Keio University.†2Faculty of Science and Technology, Keio University.†3SMBC Nikko Securities Inc. Any views or opinions
expressed in this paper are solely those of the author
and do not necessarily represent those of SMBC Nikko
Securities Inc.
Received : April 13, 2018
Accepted: March 1, 2019
of minimizing the expected cost, implicitly assum-
ing a risk-neutral investor. Almgren and Chriss [2]
derived the static optimal strategy considering the
variance of total cost as a risk measure. On the
other hand, institutional investors need to manage
the downside risk when executing the order to meet
the trading needs below a target cost. In addition,
investors develop a dynamic execution strategy and
appropriately consider the price impact cost and
market timing risk. Takenobu and Hibiki [3] for-
mulated dynamic optimal execution models with
the first-order lower partial moment (LPM) as a
downside risk measure. LPM is a practical down-
side risk measure for a security company trader to
whom the institutional investor outsources the sales
of stocks at a lower cost than the contract fee be-
cause risk can be recognized only when the total
cost exceeds a target cost. Recently, Bouchaud et
al. [4] showed that the price impact is transient in
the real market. In addition, some studies address
Vol. 70 No. 2E (2019) 105
J Jpn Ind Manage Assoc 70, 105-114, 2019
the problem under the assumption of transient MI.
Gatheral et al. [5] derived an optimal execution
strategy for minimizing the expected cost. Alfonsi
et al. [6] derived a static optimal strategy consider-
ing cost variance as a timing risk measure, which is
similar to the problem addressed by Almgren and
Chriss [2]. Lorenz and Schied [7] derived a dynamic
optimal strategy. Their admissible strategy is the
sum of the sell and buy strategies. In contrast, our
admissible execution strategy is a pure sell strategy
that aids in catering to the execution needs of in-
stitutional investors who agree to sell stocks at the
target cost.
In this paper, we discuss dynamic optimization
models assuming transient impact and downside
risk, decide the optimal execution strategy, and for-
mulate the optimal execution problem in discrete
time.
Our contributions to related literature are the
following three propositions. First, we propose
a multiperiod model with a step function (here-
after “Step model” ) involving transient MI un-
der the framework introduced by Takenobu and Hi-
biki[3]. We show that the optimal residual fraction
of shares is almost expressed in a short-butterfly
form1 in regards to cumulative cost, introducing
the effect of transient MI. Second, we propose the
following two models in order to reduce the com-
putation time. One is a multiperiod model with
a piecewise linear (PwL) function (hereafter “PwL
model”)[3] and transient MI. We find that the com-
putation time can be reduced about 70% through
PwL approximation. The other is a one-period it-
erative model (hereafter “Iterative model ” ). This
model uses a static model to solve the problem
iteratively, which drastically reduces computation
time. The state-dependent execution strategy can
be derived by solving the equation consisting of the
approximate residual fraction of shares and cumu-
lative cost using an analytical model (see Appendix
1). Third, we estimate the MI function and other
parameters using market data and derive optimal
stock execution strategies. We then show that our
1The form resembles the payoff function of the op-
tion strategy with a short position of butterfly spread,
which consists of three calls or puts with different strike
prices.
models are more useful in the real market than the
previous models. To the best of our knowledge, no
previous study has demonstrated the practicality
of model execution with transient MI using param-
eters estimated from market data. We show their
differences and characteristics in Table 1.
Table 1 Comparison of three models.
Model Step PwL Iterative
Conditional decision ⃝ ⃝ ⃝Flexibility of decision
making△ ⃝ ⃝
Inclusion of practicalconstraint
⃝ ⃝ ×
Computation load Very high High Low
Limitation of priceprocess
× △ ⃝
This paper is organized as follows. In Section
2, we briefly explain the problem, MI, and ex-
ecution cost. We also formulate a static model
(hereafter “N1 model”) with LPM, and derive the
optimal residual fraction of shares independent of
cumulative cost. In Section 3, we formulate the
Step model to derive the dynamic optimal execu-
tion strategy. We discuss the features of the state-
dependent function expressed by the Step model.
In Sections 4 and 5, we formulate the PwL model
and Iterative model, respectively. In Section 6, we
discuss the usefulness and features of the models
proposed using a sensitivity analysis. In Section
7, we show the practical application of our mod-
els through an analysis using parameters estimated
using real market data. Finally, Section 8 provides
our concluding remarks.
2 OPTIMAL EXECUTION PROBLEM
We set up the problems based on the literature
of Almgren and Chriss [2] and Alfonsi et al.[6]. We
assume that we hold a block of shares X of a single
security whose initial price is P0. In order to sell a
stock in the market by time horizon T , we divide T
into K intervals of length τ(= T/K). We then plan
to hold xk shares at time k (k = 1, . . . ,K), and sell
xk−1 − xk between k − 1 and k. The average rate
of trading during period k is νk = (xk−1 − xk)/τ .
2.1 Market Impact
In contrast to previous studies such as Almgren
and Chriss [2] and Takenobu and Hibiki [3], which
106 J Jpn Ind Manage Assoc
involve only temporary/permanent MIs as shown
on the left in the graph of Fig. 1, we take into ac-
count a transient MI, as shown on the right in the
graph of Fig. 1. We define this transient MI as fol-
lows. When τνk shares are executed, the tempo-
rary MI of h(νk) that occurs decays to the value
multiplied by Gu : [0,∞) → [0, 1], which is called
the decay kernel. Referencing Alfonsi et al.[6], we
assume Gu to be the exponential function
Gu = exp(−ρeuτ) (ρe ≥ 0)
or a power function
Gu = (1 + λuτ)−ρp (ρp, λ ≥ 0). (1)
The MI at time k from the execution at time
u− 1 (k ≥ u) is formulated as
MI((k − u+ 1)τ) = h(νu)Gk−u+1.
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Fig. 1 Temporary/permanent and transient MIs.
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Fig. 2 Decay kernels of transient MIs
We estimate the transient MI functions for the
most traded 150 stocks listed on the Tokyo Stock
Exchange using the 2012 tick data. We plot the
estimated parameters (λ, ρp) in Fig. 2. In addi-
tion, we indicate three types of parameters based
on percentiles of 25%, 50%, and 75%. We then
apply them to numerical analysis in Section 3.3.
2.2 Price Dynamics
We assume that the price process follows the
arithmetic Brownian motion.2 The evolution of
the fundamental price Pk and execution price Pk
involving MI is formulated as,
Pk = P0 + σ√τ
k∑u=1
ξu −k∑
u=1
h(νu)Gk−u+1,
Pk = Pk − h(νk).
, respectively. We represent the random price
change as σ√τξu using the daily standard de-
viation, σ, and uncertain fluctuations in period
[u− 1, u], ξu ∼ N(0, 1).
2.3 Execution Cost
Assume that a temporary MI of h(νu) occurs lin-
early against the average rate of trading νu dur-
ing period u, which is expressed as h(νu) = h0νu,
where h0 is the temporary MI coefficient. We eval-
uate the total cost of selling the amount of security
or implementation shortfall: this is the difference
between the initial market value and final capture
of the trade derived using trading policy. This is
expressed as
CK = XP0 −K∑
k=1
(xk−1 − xk)Pk
=h0
τ
K∑k=1
k∑u=1
Gk−u(xk−1 − xk)(xu−1 − xu)
−σ√τ
K−1∑k=1
ξkxk.
This can be nondimensionalized by dividing with
σ√TX, in line with Lorenz and Almgren [9] as
CK = µhK
K∑k=1
k∑u=1
Gk−u(xk−1 − xk)
×(xu−1 − xu)−K−1∑k=1
ξkxk/√K, (2)
µh = (h0X/T )/(σ√T ),
where CK = CK/(σ√TX), xk = xk/X, and µh is
a “market power” parameter, which is a “nondi-
mensional preference-free measure of portfolio size
2There is little difference between the optimal execu-
tion strategies when assuming geometric and arith-
metic Brownian motion [8]. Therefore, we use the
arithmetic Brownian motion because it is easy to han-
dle mathematically.
Vol. 70 No. 2E (2019) 107
in terms of its ability to move the market,” iden-
tified by Almgren and Lorenz [10]. The first term
of Eq. (2) shows the MI cost, and the second term
gives the timing risk. Hereafter, we remove the
caret for simplicity. Equation (2) can also be ex-
pressed as the following recurrence formula of the
cumulative cost at time k,
Ck = Ck−1 + µhK
k∑u=1
Gk−u(xk−1 − xk)
×(xu−1 − xu)− ξkxk/√K. (3)
This formula is used in Section 3.1 to formulate the
Step model.
2.4 State-independent Model (N1 Model)
We formulate a state-independent model with
downside risk based on Alfonsi et al.[6]. We use
the first-order LPM as a risk measure, which is the
expected total cost (CK) beyond the target cost
(CG). The LPM is formulated using the expected
cost (CK) and variance of cost (σ2C) as
LPM(CK) =
∫ ∞
CG
(CK − CG)g(CK)dCK
= {ϕ(Q) +QΦ(Q)}σC ,
Q = (CK − CG)/σC ,
CK = µhK
K∑k=1
k∑u=1
Gk−u(xk−1 − xk)
×(xu−1 − xu),
σ2C =
1
K
K−1∑k=1
x2k,
where CK ∼ N(CK , σ2C), g(·) is the density func-
tion of normal distribution, ϕ(·) represents the den-sity function of standard normal distribution, and
Φ(·) represents the cumulative distribution func-
tion. We formulate the N1 model for minimizing
the sum of the expected total cost and the LPM
multiplied by the risk aversion (γ ≥ 0) as follows:
min CK + γ · LPM(CK)
s.t. 1 ≥ x1 ≥ x2 ≥ · · · ≥ xK−1 ≥ 0
3 STEP MODEL
Takenobu and Hibiki [3] showed that the func-
tion form of optimal residual fraction becomes al-
most “short-butterfly” in terms of the cumulative
cost with the hybrid multi-period stochastic opti-
mization model [11] when using the LPM as a risk
measure. In this section, we show that the func-
tion form also becomes short-butterfly even for the
transient MI under the framework introduced by
Takenobu and Hibiki[3]. The Step model allows for
conditional decisions in similar states bundled each
time when using sample returns generated when
applying the Monte Carlo method, as suggested by
Hibiki [11]. We bundle the samples according to
total cost and make the same decisions for simi-
lar states. As the Step model sample, we depict a
four-period model with two nodes and four sample
paths, as shown in Fig. 3. In the next section, we
propose a PwL model that uses the short-butterfly
function form to obtain the optimal residual frac-
tion of shares in order to reduce the total compu-
tation time based on the Step model which gives
a dynamic optimal strategy with the step function
form. Therefore, our discussion begins with the
Step model to understand the PwL model.
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Fig. 3 Structure of the Step model.
3.1 Step Model Formulation
We formulate a Step model that uses S nodes
(hereafter “NS model”) for the optimal execution
problem.
(1) Notations
a) Parameters
J : number of sample paths (j = 1, . . . , J)
K: number of periods (k = 1, . . . ,K)
S: number of nodes (s = 1, . . . , S)
ξ(j)k : random price change on path j at time k
µh: market power
108 J Jpn Ind Manage Assoc
γ: risk aversion coefficient
CG: target cost
b) Variables
x1: residual fraction of order held at time 1, de-
termined at time 0
ysk: residual fraction held on node s at time k
q(j): deviation of total cost CK above CG on path
j
x(j)k : residual fraction held on path j at time k
C(j)k : cumulative cost on path j up to time k
LPM(CK): first-order LPM of total cost
(2) Formulation (Description of (j = 1, . . . , J) is
omitted in the constraints involving the super-
script (j))
Minimize1
J
J∑j=1
C(j)K + γ · LPM(CK)
subject to
C(j)k = C
(j)k−1 + µhK
k∑u=1
Gk−u(x(j)k−1 − x
(j)k )
×(x(j)u−1 − x(j)
u )− ξ(j)k x
(j)k /√K (k = 1, . . . ,K)
(C(j)0 = 0, x
(j)0 = 1, x
(j)1 = x1, x
(j)K = 0) (4)
C(j)K − q(j) ≤ CG (5)
LPM(CK) =1
J
J∑j=1
q(j) (6)
q(j) ≥ 0 (7)
x(j)k ≤ x
(j)k−1 (k = 1, . . . ,K) (8)
x(j)k =
y1k (C
(j)k−1 ≤ θ1k−1)
ysk (θs−1
k−1 ≤ C(j)k−1 ≤ θsk−1,
s = 2, . . . , S − 1)
ySk (C
(j)k−1 ≥ θS−1
k−1 )
(9)
(k = 2, . . . ,K − 1)
Equation (4) is the calculation of cumulative cost
up to time k. Equations (5) to (7) are used for
calculating the LPM. Equation (8) is the non-
increasing constraint of time for residual fractions.
Equation (9) gives the residual fraction of order
x(j)k , which is the step function of cumulative cost
C(j)k−1; the residual fractions ys
k are determined on
node s. θsk−1 gives the portioned points of C(j)k−1 on
node s. Conditional decisions can be made in the
model.
3.2 Iterative Algorithm for Optimization
The Step model is formulated as a non-convex
and non-linear programming problem. This is be-
cause the optimal residual fractions are determined
using the cumulative costs of the portioned points,
but the cumulative costs depend on the residual
fractions. Therefore, we can derive an approxi-
mate solution using an iterative algorithm, as well
as multi-period stochastic optimization introduced
by Hibiki[11] as follows.
(Step 1) We derive the optimal static residual or-
ders using the N1 model. Set m = 2;
(Step 2) We calculate the (m − 1)-th cumulative
cost C(j)∗k−1(m−1) and determine the (m − 1)-th
threshold θs∗k−1(m−1);
(Step 3) We derive the optimal residual fraction of
node s at time k (ysk) using the Step model with
C(j)∗k−1(m−1) and θs∗k−1(m−1) as parameters; and
(Step 4) Stop if the difference between the m-th
and the (m− 1)-th objective function values is
lower than the tolerance level. Otherwise, set
m← m+ 1 and return to Step 2.
3.3 Optimal Execution Strategy and Step
Function
We estimate a step function through numerical
analysis of a Step model with many nodes using the
above algorithm. We set the following parameters:
J = 50, 000, K = 6, S = 25, γ = 1, µh = 0.1, and
CG = 0.3. We assume a transient MI as the power
function with λ = 50, and ρp = 0.25 in Eq. (1) as
the base case with reference to the empirical re-
sult in Section 2.1. We derive the dynamic optimal
execution strategy using the N25 Step model, and
bundle the same number of paths according to the
total cost in each range (J/S).
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Fig. 4 Dynamic optimal execution strategy.
Vol. 70 No. 2E (2019) 109
We show the strategy in Fig. 4. The optimal
residual orders are dependent on the cumulative
cost, and the state-dependent functions almost take
short-butterfly forms, consisting of a V-shaped part
and flatter parts. As the cumulative cost becomes
close to the kinked point, the chance of risk be-
comes large. Therefore, the MI is tolerated and the
amount of residual order becomes small to avoid
the increase in timing risk, which is difficult to con-
trol. On the other hand, the residual order to have
a chance of reducing the total cost as the result
of rising stock price becomes large as the cumu-
lative cost becomes larger than the kinked point.
The introduction of a transient MI increases the
executed fractions in the latter and earlier peri-
ods, and the effects of cost reduction due to the
decay of MI. We compare the optimal execution
strategies for two cases of transient MI with refer-
ence to Fig. 2: (λ, ρp) = (25, 0.35) for Case 1, and
(λ, ρp) = (125, 0.17) for Case 2. We use the dy-
namic optimal execution strategies shown on the
left-hand side of Fig. 5 and the average amount of
the fraction at each execution time on the right-
hand side. We show that the results are consistent
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Fig. 5 Optimal execution strategies for two cases
with Takenobu and Hibiki[3], even for a transient
MI, demonstrating the effect of transient MI. As
expected, the effect is larger for Case 1 than Case 2
because the decay kernel of Case 1 is smaller than
that of Case 2. We need a large number of nodes
to express the short-butterfly form using the step
function in the Step model. To reduce the compu-
tation time and solve the large-scale optimization
problem, we propose the PwL model with the tran-
sient MI, as reported in Takenobu and Hibiki[3].
The computation time can be reduced using the
PwL approximation shown in Section 4. In addi-
tion, we propose a method to solve the problem
using the Iterative model to reduce computation
time drastically, as shown in Section 5.
4 PwL MODEL
The cumulative cost to the lowest fraction needs
to be given in the PwL model before solving the
problem (see Fig. 6). However, it is difficult to
determine this because we cannot find it without
solving the problem using the Step model. In this
study, we determine this as CG− (Ak +Bk), or the
“target cost minus the sum of expected cumulative
cost after time k and risk adjusted term,” derived
using the one-period analytical model formulated
under the simplified conditions shown in Appendix
1. Ak and Bk are calculated as
Ak = CK − Ck−1,
Bk = bkxmink v∗,
bk =1
xmink
√√√√ 1
K
K−1∑t=k
(xmint )2,
v∗ =ϕ(v∗)
1 + 1/γ − Φ(v∗), (10)
where xmink is a residual fraction of order in the
kinked point derived using the analytical model.
Equation (10) is solved for v, and v∗ is derived.
We plot several combinations, using both the val-
ues of CG− (Ak +Bk) on the horizontal x-axis and
the cumulative costs to the lowest fraction in the
Step model on the vertical y-axis, in Fig. 7 to ex-
amine their relationship. We find that this is the
appropriate expression because these plots are near
the 45-degree line.
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Fig. 6 Function forms for two models.
We formulate the PwL model with S ranges
(hereafter “PS model”) as an extension of the Step
model. We also use the parameters and variables
110 J Jpn Ind Manage Assoc
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3
Kin
ked
poin
t b
y S
tep
mod
el
Kinked point : �_�−�_�−�_�
gamma=1gamma=2gamma=3gamma=4gamma=5gamma=10
Fig. 7 Examination of minimum kinked point.
defined in Section 3.1. The formulation of the PwL
model is basically the same as that of the Step
model, except that the calculation of x(j)k in Eq.(9)
is replaced in Eq.(11):
x(j)k =
y1k (C
(j)k−1 ≤ θ1k−1)
(1− α(j)k )ys−1
k + α(j)k ys
k (θs−1k−1 ≤
C(j)k−1 ≤ θsk−1, s = 2, . . . , S − 1)
ySk (C
(j)k−1 ≥ θS−1
k−1 )
(11)
α(j)k =
C(j)k−1 − θs−1
k−1
θsk−1 − θs−1k−1
, (k = 2, . . . ,K − 1)
The iterative algorithm in Section 3.2 can be ap-
plied for optimization. We decide the optimal resid-
ual orders continuously using Eq. (11). A similar
optimal execution strategy can be derived using the
PwL model with a smaller number of ranges than
the Step model.
5 ITERATIVE MODEL
Even when using the PwL model, the computa-
tion time increases in proportion to the number of
periods (K), as with the Step model. We propose
the Iterative model to overcome the computation
problem. Furthermore, this model can make state-
dependent decisions similar to those when using the
PwL model.
Consider the problem of solving for xk. Given
Ck−1 and xk−1, we develop an equation for the rela-
tionship between xk and Ck−1, as in Eq. (12), using
the analytical model (see Appendix 1).
a1k
a2k− xk =
bkh(v)
2µhKa2k(12)
where v is a function of Ck−1 and xk(see Eq. (19)),
and a1k and a2k are functions of xk−1 but not
xk(see Eqs. (22) and (23)). Equation (12) cannot
be solved analytically, but given xk−1, we obtain
xk numerically with respect to various Ck−1, as
shown in Fig. 8. The optimal residual fractions de-
rived from Eq. (12) are dependent on the cumula-
tive cost, and the state-dependent functions almost
take short-butterfly forms.
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Fig. 8 One-period dynamic execution strategy using
the analytical model.
We derive approximately the state-dependent
residual orders (x2, . . . , xK−1) by implementing the
following procedure (see Appendix 2 for details).
(Step 1) We derive the static residual orders
xN1k (k = 1, . . . ,K−1) using the N1 model. The
optimal execution strategy with the Iterative
model is denoted by x∗k hereafter;
(Step 2) We set x∗1 = xN1
1 and k = 2;
(Step 3) We calculate Ck−1 using x∗k−1 and bk
using xN1t (t ≥ k), and determine x∗
k using
Eq. (12); and
(Step 4) We perform Step 3 for k = 3, ...,K − 1.
This procedure enables us to derive the state-
dependent residual orders in considerably less time
because the computation for solving the problem
using the N1 model takes only a few seconds and
the other calculations are also very simple. Next,
we numerically analyze the two models proposed in
Sections 4 and 5.
6 NUMERICAL ANALYSIS
We derive the optimal execution strategy with
the PwL and Iterative models using hypotheti-
cal data, compare the results of the Step model,
and conduct a sensitivity analysis to examine the
usefulness of the models. All the problems are
solved using the trust region method with Numeri-
cal Optimizer (Ver 18.1), a mathematical program-
ming software package developed by NTT DATA
Mathematical System, Inc., on a personal computer
equipped with the Windows 10 OS, Core i7-6700K,
a 4.00 GHz CPU and 32 GB of memory.
Vol. 70 No. 2E (2019) 111
6.1 Setting
(1) Parameters of base case
The parameters are: K=6, J=50,000, CG=0.3,
γ=1, µh=0.1, λ = 50, and ρp=0.25.
(2) How to classify paths
In the Step and PwL models, we set eight types
of S (=2, 4, 6, 8, 12, 16, 20, 24) and eight ranges
at each time. For the PwL model, the number of
divisions is symmetric to the kinked point, which
is the cumulative cost to the minimum fraction
(CG−(Ak+Bk)) estimated in Section 4. We set αJ
paths (α=0.04) in the flatter parts at both ends to
express short-butterfly forms, as done by Takenobu
and Hibiki[3].
6.2 Basic Analysis
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Fig. 9 Optimal execution strategies.
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Fig. 10 Objective function values for different num-
bers of divisions.
We show the optimal execution strategies in
Fig. 9, derived by the P8 and Iterative models in
the base case. The optimal strategies are similar to
those of the Step model. In addition, we show the
objective function values for different numbers of
divisions in Fig. 10. When S=1, the Step and PwL
models are equivalent to the N1 model. We show
the constant objective value for the Iterative model
because it has no relation with the number of di-
visions. By comparing the N16 and P6 or the N24
and P12 models, which have similar objective func-
tion values,3 the computation time can be reduced
about 70%.
6.3 Sensitivity Analysis
We conduct a sensitivity analysis to compare four
models for different parameters of γ and ρp. Fig-
ure 11 shows the difference in the objective function
values for the N24 (Step) model. The PwL and It-
erative models are close to the Step model. Fur-
thermore, the PwL model is more accurate than
the Iterative model and robust to changes under
the market conditions.
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Fig. 11 Sensitivity analysis results.
7 ANALYSIS OF MARKET DATA
We estimate the MI function and other param-
eters using 2012 tick data, and derive the optimal
execution strategies for practical use. Specifically,
we estimate the temporary MI coefficient h0 as the
spread over best bid quality before the day of ex-
ecution, the daily standard deviation σ as the re-
alized volatility, and the transient MI function Gu
referencing the method of Bouchaud et al.[4]. We
calculate the market power µh using h0 and σ, and
set the amount of X based on the average trading
volume per day.
Table 2 Parameters estimated using market data.
Parameter Softbank (9984) SQUARE ENIX (9684)
P0 3,140 yen 1,095 yen
σ 26.68 yen 16.81 yen
h0 2.5 ×10−5 yen 6.6 ×10−4 yen
Gu (1 + 179.03uτ)−0.26 (1 + 3.09uτ)−1.23
X 800,000 80,000
3We confirm that the optimal execution strategies of the
N16 and P6 models and those of the N24 and P12 mod-
els are similar.
112 J Jpn Ind Manage Assoc
Table 3 Comparison of models; differences in objec-
tive function values from N25 model (Dif.)
and computation time (min).
9984 9684
Models Dif. Min Dif. Min
N25 model 0.00% 199.9 0.00% 171.4
P8 model 0.01% 69.4 0.00% 54.8
Iterative model 0.14% 0.0 0.05% 0.0
N1 model 0.65% 0.0 0.05% 0.0
Equally model 6.25% 0.0 1.72% 0.0
Takenobu and Hibiki[3] 0.11% 68.1 1.05% 53.1
Suppose that the shares of Softbank (9984: secu-
rity code) and SQUARE ENIX (9684), which are
large-scale stocks listed on the first section of the
Tokyo Stock Exchange, are executed and the esti-
mated parameters are as shown in Table 2. The
transient MI decay speed of Softbank is fast and
that of SQUARE ENIX is slow. We derive the
optimal execution strategies using the three mod-
els proposed (N25 (Step), P8 (PwL), and Iterative)
and three other models (execution strategy of trad-
ing in equal lot size (Equally model), N1 model, and
P8 model with temporary/permanent MI as pro-
posed by Takenobu and Hibiki[3]). We compare the
objective function values with the N25 model and
show the total computation time of each model in
Table. 3. The objective function values of the three
models proposed are smaller than those of the other
three models. Especially when executing SQUARE
ENIX, whose transient MI decays very slowly, the
improvement of the objective function value of the
proposed P8 model is prominent compared to PwL
using the temporary/permanent model [3]. Fur-
thermore, the computation time is reduced when
using the PwL and Iterative models. This implies
that the proposed models are useful in practice.
8 CONCLUSION
We proposed three different types of models for
dynamic optimal execution models: Step, PwL,
and Iterative. We showed the characteristics and
usefulness of each model using the state-dependent
strategy with hypothetical and real market data.
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APPENDIX
1 Analytical model used in the PwL model
Under the simplified conditions with respect to
constraints, we solve the optimal execution prob-
lem analytically and derive an equation expressing
the relationship xk and Ck−1 to show the method
of deriving the kinked point in Section 4. Consider
the problem of solving for xk. Given Ck−1 and
xk−1, the expected total cost (CK) and variance of
total cost (σ2C) are formulated using the residual
orders after time k, xt (t = k + 1,…,K − 1) as
CK = Ck−1 + µhK∑K
t=k
∑tu=1 Gt−u
Vol. 70 No. 2E (2019) 113
×(xt−1 − xt)(xu−1 − xu), (13)
σ2C = 1
K
∑Kt=k x
2t . (14)
Suppose that the residual fraction of order at
time t (t = k, ...,K − 1) is xt = bktxk. Equations
(13) and (14) can be expressed as,
CK = Ck−1 + µhK{(β4k − β3k + 1)x2k
+(β2k + β3kxk−1 − 2xk−1 − β1k)xk
+(x2k−1 + β1kxk−1)},
σC = bkxk,
where
bk =√
1K
∑K−1t=k b2kt,
β1k =∑k−1
u=1 Gk−u(xu−1 − xu), (15)
β2k =∑K
t=k+1
∑k−1u=1 Gt−u(bk,t−1 − bkt)
×(xu−1 − xu), (16)
β3k =∑K
t=k+1 Gt−k(bk,t−1 − bkt), (17)
β4k =∑K
t=k+1
∑tu=k+1 Gt−u(bk,t−1 − bkt)
×(bk,u−1 − bku). (18)
The objective function can be expressed as
f(xk) = CK + γLPM(CK)
= CG +{v(γΦ(−v)− 1) + γϕ(v)
}σC ,
v = (CG − CK)/σC . (19)
We derive xk by minimizing the objective func-
tion f(xk) and satisfying f ′(xk) = 0, and develop
Eq. (20).
a1ka2k− xk = bkh(v)
2µhKa2k(20)
h(v) = ϕ(v)Φ(−v)+1/γ
(21)
a1k = xk−1 +12β1k − 1
2β2k − 1
2β3kxk−1 (22)
a2k = β4k − β3k + 1 (23)
Equation (20) represents the equation between
xk and Ck−1, and therefore holds for the minimum
fraction of xk. We solve the problem to minimize
xk and find the kinked point (C∗k−1, x
∗k) referencing
Takenobu and Hibiki[3] as follows.
(1) We solve h′(v) = 0, and obtain v∗ for max-
imizing h(v). The solution is satisfied with
v∗ = h(v∗) and calculated numerically. v∗ de-
pends on γ only.
(2) x∗k is calculated by substituting v∗ into
Eq. (20).
(3) We define A∗k = CK − C∗
k−1. Because CK =
CG− q∗ and q∗ = σ∗Cv
∗ = bkx∗kv
∗, C∗k−1 can be
calculated as
C∗k−1 = CG − (A∗
k +B∗k), where B∗
k = bkx∗kv
∗.
(24)
We then calculate A∗k as the expected total cost af-
ter time k, and derive the cumulative cost to the
kinked point as shown in Eq. (24).
2 Iterative model
First, we compute the residual fractions of order
at time k, zk(k = 1, . . . ,K−1) using the N1 model.
We derive the combinations of(C
(j)k−1, x
(j)∗k
)for
k = 2, . . . ,K − 1 through a procedure sequentially
using Eqs. (25) to (28)(in 1⃝ to 4⃝, respectively) as
follows:
k = 1 : x∗1 = z∗1
2⃝→ C
(j)∗1
k = 2 : (z∗1 , C(j)∗1 )
3⃝→C
(j)∗K
4⃝→ v(j)
1⃝⇒x
(j)∗2
2⃝→C
(j)∗2
k = 3 . . . ,K − 1 : (z∗k−1, C(j)∗k−1)
3⃝→ C
(j)∗K
4⃝→ v(j)
1⃝⇒ x
(j)∗k
2⃝→ C
(j)∗k
When we take a pure sell strategy, we need to sat-
isfy the constraints x(j)k ≤ x
(j)k−1 (k = 2, . . . ,K).
However, since we cannot involve it explicitly in
this procedure, we force x(j)k = x
(j)k−1 if the con-
straints are not satisfied. The equations used in
the procedure are as follows.
1⃝ An equation expressing the relationship be-
tween cumulative cost and the residual fraction
of order, which is equivalent to Eq. (20)
x(j)k = 1
a2k
(a1k − bkh(v
(j))2µhK
)(25)
where h(v), a1k, and a2k are calculated using
Eqs. (21) to (23), and βki(i = 1, 2, 3, 4) are
constants calculated in Eqs. (15) to (18).
2⃝ Calculating the cumulative cost up to time k,
C(j)k = C
(j)k−1 + µhK
∑ku=1 Gk−u(x
(j)k−1 − x
(j)k )
×(x(j)u−1 − x(j)
u )− x(j)k ξ
(j)k /√K. (26)
3⃝ Calculating the expected total cumulative cost,
C(j)K = C
(j)k−1 + µK
[(β4k − β3k + 1)(x
(j)k )2
+ {β2k + (β3k − 2)zk−1 − β1k}x(j)k
+(z2k−1 + β1kzk−1)]. (27)
4⃝ Calculating the normalized target costs,
v(j) =CG−C
(j)K
σ(j)C
, σ(j)C = bkx
(j)k . (28)
114 J Jpn Ind Manage Assoc