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


×(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


×(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


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


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.


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.

REFERENCES

[1] Bertsimas, D. and Lo, A.: “Optimal Control

of Execution Costs,” J. Financ. Mark., Vol. 1,

pp. 1–50 (1998)

[2] Almgren R. and Chriss, N.: “Optimal Execution

of Portfolio Transactions,” J. Risk, Vol. 3, pp. 5–

40 (2001)

[3] Takenobu, S. and Hibiki, N.: “Dynamic Optimal

Execution Model with Downside Risk,” Com-

mun. Oper. Res. Jpn., Vol. 61, No. 6, pp. 384–

395 (2016) (in Japanese)

[4] Bouchaud, J.P., Kockelkoren, J. and Potters, M.:

“Random Walks, Liquidity Molasses and Critical

Response in Financial Markets,” Int. J. Theor.

Appl. Financ., Vol. 14, No. 3, pp. 353–368 (2011)

[5] Gatheral, J., Schied A. and Slynko, A.: “Tran-

sient Linear Price Impact and Fredholm Inte-

gral Equations,” Math. Financ., Vol. 22, No. 3,

pp. 445–474 (2011)

[6] Alfonsi,A., Schied, A. and Slynko, A.: “Order

Book Resilience, Price Manipulation, and the

Positive Portfolio Problem,” SIAM J. Financ.

Math., Vol. 3, No. 1, pp. 511–533 (2012)

[7] Lorenz, C. and Schied, A.: “Drift Dependence

of Optimal Trade Execution Strategies under

Transient Price Impact,” Financ. Stoch., Vol. 17,

No. 4, pp. 743–770 (2013)

[8] Gatheral, J. and Schied, A.: “Optimal Trade

Execution under Geometric Brownian Motion in

the Almgren and Chriss Framework,” Math. Fi-

nanc., Vol. 22, No. 3, pp. 445–474 (2011)

[9] Lorenz, J. and Almgren, R.: “Mean-Variance

Optimal Adaptive Execution,” Appl. Math. Fi-

nanc., Vol. 18, No. 5, pp. 395–422 (2011)

[10] Almgren, R. and Lorenz, J.: “Adaptive Arrival

Price,” J. Trading, Vol. 1, pp. 59–66 (2007)

[11] Hibiki, N.: “Multi-period Stochastic Optimiza-

tion Models for Dynamic Asset Allocation,” J.

Bank. Financ., Vol. 30, No. 2, pp. 365–390 (2006)

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)


dynamic optimal execution models with transient market

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