a study on optimal execution strategy model with limit ...3 optimal execution strategy model 31 ......
TRANSCRIPT
Master Thesis Academic Year: 2016-2018
A Study on Optimal Execution StrategyModel with Limit Orders Using
Multi-Period Stochastic ProgrammingInvolving Unexecution Risk
Shumpei SakuraiStudent ID: 876008
SUPERVISOR: Professor Mosconi Rocco Robert
CO-SUPERVISOR: Professor Norio Hibiki (Keio University)
2018/04
Politecnico di Milano
Scuola di Ingegneria Industriale e dell’Informazione
Ingegneria Gestionale
Contents
1 Introduction 16
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2 Objective of this Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Basic Concepts 22
2.1 Market Order and Limit Order . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Market Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Market Impact of Market Order . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Market Impact of Limit Order . . . . . . . . . . . . . . . . . . . . . 25
2.3 Timing Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Unexecution Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Three Risks and Order Types . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Optimal Execution Strategy Model 31
3.1 The Limit Order Book and Two Order Types . . . . . . . . . . . . . . . . . . 31
3.2 Literatures about Limit Order Book . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Literatures about Market Order Strategy . . . . . . . . . . . . . . . . . . . . 33
3.4 Literatures about Limit Order Strategy . . . . . . . . . . . . . . . . . . . . . 38
3.5 Literatures about Limit Order Book Strategy . . . . . . . . . . . . . . . . . . 44
3.6 Novelties of this Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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3.7 Objectives, Research Methodology and Research Framework . . . . . . . . . 48
3.7.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.7.2 The Simulation Multi-Period Stochastic Programming . . . . . . . . 51
3.7.3 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7.4 Constraints of Executed Order Volume . . . . . . . . . . . . . . . . 55
3.7.5 Execution Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7.6 Price Dynamics and Market Impact Risk . . . . . . . . . . . . . . . 59
3.7.7 Constraints of Execution Cost . . . . . . . . . . . . . . . . . . . . . 61
3.7.8 Constraints of Target Executed Volume . . . . . . . . . . . . . . . . 61
3.7.9 Constraints ofxt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Parameter Estimation 63
4.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Cont et al. [2013] (Estimation of Market Impact) . . . . . . . . . . . 63
4.1.2 Omura et al. [2000] (Estimation of Execution probability) . . . . . . 65
4.1.3 Lo et al. [2002] (Estimation of Execution Probability) . . . . . . . . 67
4.1.4 Cont-Kukanov [2017] (Estimation of Execution Probability) . . . . . 68
4.2 Objectives, Research Methodology, and Research Framework . . . . . . . . . 71
4.2.1 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.2 Estimation of Market Impact . . . . . . . . . . . . . . . . . . . . . . 74
4.2.3 Estimation of Price Volatility . . . . . . . . . . . . . . . . . . . . . . 76
4.2.4 Estimation of Execution Probability . . . . . . . . . . . . . . . . . . 77
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4.3 Findings, Discussion, and Conclusions . . . . . . . . . . . . . . . . . . . . . 81
4.3.1 Sum of Order Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.2 Average Order Size . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.3 Standard Deviation of Order Size . . . . . . . . . . . . . . . . . . . 89
4.3.4 Order Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.5 The Number of Orders . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3.6 Estimation of Price Volatilityσ . . . . . . . . . . . . . . . . . . . . 104
4.3.7 The Order Flow Imbalance and Price Change . . . . . . . . . . . . . 108
4.3.8 Estimation of Market Impact: Model 0 . . . . . . . . . . . . . . . . 113
4.3.9 Estimation of Market Impact: Model 1 . . . . . . . . . . . . . . . . 117
4.3.10 Estimation of Market Impact: Model 2 . . . . . . . . . . . . . . . . 123
4.3.11 Estimation of Execution Probability . . . . . . . . . . . . . . . . . . 131
4.3.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Application of the Optimal Strategy Model 135
5.1 Basic Parameter Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 Sensitivity of the Optimal Strategy to Parameter Settings . . . . . . . . . . . 140
5.2.1 Relative Percentage of Target Order Value . . . . . . . . . . . . . . . 140
5.2.2 Confidence Interval of CVaRβ . . . . . . . . . . . . . . . . . . . . 145
5.2.3 Risk Aversion Coefficientγ . . . . . . . . . . . . . . . . . . . . . . 151
5.2.4 Number of PeriodsN . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.2.5 Market Impact of Limit OrderLMI . . . . . . . . . . . . . . . . . . 157
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5.2.6 Market Impact of Market OrderMMI . . . . . . . . . . . . . . . . . 161
5.2.7 Mean of Execution Probability E[τ] . . . . . . . . . . . . . . . . . . 164
5.2.8 Standard Deviation of Execution Probability sd[τ] . . . . . . . . . . 168
5.2.9 Volatility of Price Changeσ . . . . . . . . . . . . . . . . . . . . . . 173
5.3 Another Model Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.3.1 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.3.2 Market Order Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.3.3 Target of Market Impact . . . . . . . . . . . . . . . . . . . . . . . . 185
5.3.4 Strategy without Reorder . . . . . . . . . . . . . . . . . . . . . . . . 187
6 Conclusions 189
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List of Figures
1 Dynamics of Deposit Facility Rate in ECB . . . . . . . . . . . . . . . . . . . 16
2 Average Life Expectancy of Japan (Source: Cabinet Office of Japan [2017]) . 17
3 Market Order without Market Impact . . . . . . . . . . . . . . . . . . . . . . 24
4 Market Order with Market Impact . . . . . . . . . . . . . . . . . . . . . . . 25
5 Limit Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Patterns of Market Impact of Limit Order (LMI) . . . . . . . . . . . . . . . . 27
7 Simulation Multi-Period Stochastic Programming Model . . . . . . . . . . . 51
8 Image of Cumulative Cost Distribution and CVaR . . . . . . . . . . . . . . . 54
9 Image of the Execution Strategy . . . . . . . . . . . . . . . . . . . . . . . . 57
10 Probability Distributions of Execution Volume . . . . . . . . . . . . . . . . . 59
11 Filled Amount in Cont-Kukanov [2017] Model . . . . . . . . . . . . . . . . 69
12 Relative Percentage of OFI Elements to Depth (All Stocks) . . . . . . . . . . 79
13 Boxplot of Sum of Order Size (Stock A, Interval=15min) . . . . . . . . . . . 83
14 Medians of Sum of Order Size (All Stocks, Interval=15min) . . . . . . . . . 85
15 Distribution of Average Order Size (Stock A, Interval=15min) . . . . . . . . 87
16 Distribution of Average Order Size (All Stocks, Interval=15min) . . . . . . . 89
17 Distribution of Standard Deviation of Order Size (Stovk A, Interval=15min) . 91
18 Distribution of Standard Deviation of Order Size (All Stocks, Interval=15min) 92
19 Distribution of Order Value (Stock A, Interval=15min) . . . . . . . . . . . . 94
20 Distribution of Order Value (All Stocks, Interval=15min) . . . . . . . . . . . 96
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21 Distribution of the Number of Orders (Stock A, Interval=15min) . . . . . . . 98
22 Distribution of Order Volume without Outliers (Stock A, Interval=15min) . . 100
23 Distribution of Order Volume (All Stocks, Interval=15min) . . . . . . . . . . 103
24 Price Dynamics (Stock A) . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
25 Price Dynamics (Stock B) . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
26 Price Dynamics (Stock C) . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
27 Price Dynamics (Stock D) . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
28 Absolute Price Change and OFI (Stock A, Absolute Price Difference) . . . . 108
29 Percentage Price Change and OFI (Stock A, Percentage Price Difference) . . 109
30 Log Price Change and OFI (Stock A, Log Price Difference) . . . . . . . . . . 110
31 Absolute Price Change and TI (Stock A) . . . . . . . . . . . . . . . . . . . . 111
32 Absolute Price Change and OFIWOTI (Stock A) . . . . . . . . . . . . . . . 112
33 Estimated Execution Probability (All Stocks) . . . . . . . . . . . . . . . . . 132
34 Posted Order Unit of a set of Simulation Paths . . . . . . . . . . . . . . . . . 136
35 Average Posted Order Unit of Four Stocks . . . . . . . . . . . . . . . . . . . 137
36 Execution Cost of Four Stocks . . . . . . . . . . . . . . . . . . . . . . . . . 139
37 Average Posted Order Value under Target Volume Settings . . . . . . . . . . 142
38 Average Posted Order Unit under Target Volume Settings . . . . . . . . . . . 143
39 Execution Cost under Target Volume Settings . . . . . . . . . . . . . . . . . 144
40 Average Posted Order Unit underβ Settings (Case-0.552%) . . . . . . . . . 146
41 Average Posted Order Unit underβ Settings (Case-55.187%) . . . . . . . . . 146
42 Distribution of Execution Cost underβ Settings (Case-55.187%) . . . . . . . 148
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43 Execution Cost underβ Settings (Case-0.552%) . . . . . . . . . . . . . . . . 149
44 Execution Cost underβ Settings (Case-55.187%) . . . . . . . . . . . . . . . 150
45 Average Posted Order Unit underγ Settings (Case-0.552%) . . . . . . . . . . 151
46 Average Posted Order Unit underγ Settings (Case-55.187%) . . . . . . . . . 152
47 Execution Cost underγ Settings (Case-0.552%) . . . . . . . . . . . . . . . . 153
48 Execution Cost underγ Settings (Case-55.187%) . . . . . . . . . . . . . . . 153
49 Average Unexecuted Order Unit under N Settings . . . . . . . . . . . . . . . 155
50 Execution Cost under N Settings . . . . . . . . . . . . . . . . . . . . . . . . 156
51 Average Posted Order Unit under LMI Settings (Case-0.552%) . . . . . . . . 157
52 Average Posted Order Unit under LMI Settings (Case-55.187%) . . . . . . . 158
53 Execution Cost underLMI Settings (Case-0.552%) . . . . . . . . . . . . . . 159
54 Execution Cost underLMI Settings (Case-55.187%) . . . . . . . . . . . . . 159
55 Average Posted Order Unit under MMI Settings (Case-0.552%) . . . . . . . 161
56 Average Posted Order Unit under MMI Settings (Case-55.187%) . . . . . . . 161
57 Execution Cost under MMI Settings (Case-0.552%) . . . . . . . . . . . . . . 162
58 Execution Cost under MMI Settings (Case-55.187%) . . . . . . . . . . . . . 163
59 Average Posted Order Unit under E[τ] Settings (Case-0.552%) . . . . . . . . 164
60 Average Posted Order Unit under E[τ] Settings (Case-55.187%) . . . . . . . 165
61 Execution Cost under E[τ] Settings (Case-0.552%) . . . . . . . . . . . . . . 166
62 Execution Cost under E[τ] Settings (Case-55.187%) . . . . . . . . . . . . . 166
63 Average Posted Order Unit under sd[τ] Settings (Case-0.552%) . . . . . . . . 168
64 Average Posted Order Unit under sd[τ] Settings (Case-55.187%) . . . . . . . 169
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65 Standard Deviation of ND and TND . . . . . . . . . . . . . . . . . . . . . . 170
66 Execution Cost under sd[τ] Settings (Case-0.552%) . . . . . . . . . . . . . . 171
67 Execution Cost under sd[τ] Settings (Case-55.182%) . . . . . . . . . . . . . 172
68 Average Posted Order Unit underσ Settings (Case-0.552%) . . . . . . . . . 173
69 Average Posted Order Unit underσ Settings (Case-55.187%) . . . . . . . . . 174
70 Execution Cost underσ Settings (Case-0.552%) . . . . . . . . . . . . . . . . 175
71 Execution Cost underσ Settings (Case-55.187%) . . . . . . . . . . . . . . . 176
72 Average Posted Order Unit under Two Distributions (Case-55.187%) . . . . . 177
73 Standard Deviation of Binomial and TN Distribution . . . . . . . . . . . . . 180
74 Execution Cost of Two Distributions . . . . . . . . . . . . . . . . . . . . . . 181
75 Average Posted Order Unit of Market Order and Limit Order Strategy . . . . 182
76 Execution Cost of Market Order Strategy . . . . . . . . . . . . . . . . . . . 183
77 Average Posted Order under Two Settings of Market Impact . . . . . . . . . 185
78 Average Posted Order Unit of Market Order and Limit Order Strategy . . . . 187
79 Execution Cost of Reorder and Without Reorder Strategies . . . . . . . . . . 188
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List of Tables
1 Main Types of Traders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Order Types and Their Risks . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Comparison with Previous Researches . . . . . . . . . . . . . . . . . . . . . 47
4 Example of CVaR Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 Four Assets and Their Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Relative Volume of OFI Elements to Depth (Stock A, 8306) . . . . . . . . . . 78
7 Distribution of Sum of Order Size (Stock A, Interval=15min) . . . . . . . . . 82
8 Correlation Matrix of Sum of Order Size (Stock A, Interval=15min) . . . . . 84
9 Distribution of Average Order Size (Stock A, Interval=15min) . . . . . . . . 86
10 Correlation of Average Order Size (Stock A, Interval=15min) . . . . . . . . . 88
11 Distribution of Standard Deviation of Order Size (Stock A, Interval=15min) . 90
12 Distribution of Order Value (Stock A, Interval=15min) . . . . . . . . . . . . 93
13 Correlation Matrix of Order Value (Stock A, Interval=15min) . . . . . . . . . 95
14 Distribution of the Number of Orders (Stock A, Interval=15min) . . . . . . . 97
15 Distribution of Number of Orders without Outliers (Stock A, Interval=15min) 99
16 Correlation Matrix of Number of Orders(Stock A, Interval=15min) . . . . . . 101
17 Correlation Matrix of Number of Orders without Outliers (Stock A, Inter-
val=15min) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
18 Result of Price Volatilityσ Estimation . . . . . . . . . . . . . . . . . . . . . 107
19 Result of Model 0 Regression (per 1 million yen, Absolute Price Difference) . 113
9
20 Other Results of Model 0 (Absolute Price Difference) . . . . . . . . . . . . . 114
21 Result of Model 0 Regression (per 1 million yen, Percentage Price Difference) 115
22 Result of Model 0 Regression (per 1 million yen, Log Price Difference) . . . 116
23 Model 1 - Log Price Difference and Order Value (Stock A) . . . . . . . . . . 117
24 Model 1 - Log Price Difference and Order Value (All Stocks, Coefficients) . . 118
25 Other Results of Model 1 (All Stocks, Coefficients) . . . . . . . . . . . . . . 119
26 Model 1 - Log Price Difference and Value Test (All Stocks) . . . . . . . . . . 121
27 VIFs of Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
28 Top 10 Assets of Portfolio Holdings as of March 31, 2017 (Source: GPIF
[2017]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
29 Relative Importance of 100 Million yen . . . . . . . . . . . . . . . . . . . . 126
30 Relationship among Value, Volume, and Unit . . . . . . . . . . . . . . . . . 127
31 Result of Regression (per Unit, Percentage Price Difference) . . . . . . . . . 128
32 Other Result of Regression (per Unit, Percentage Price Difference) . . . . . . 129
33 VIF of Model2 - No Cancel Order . . . . . . . . . . . . . . . . . . . . . . . 130
34 Estimated Execution Probability (New Model) . . . . . . . . . . . . . . . . . 131
35 Estimated Execution Probability (Old Model) . . . . . . . . . . . . . . . . . 132
36 Basic Estimated Parameter Settings . . . . . . . . . . . . . . . . . . . . . . 134
37 Relationship between LMI and MMI . . . . . . . . . . . . . . . . . . . . . . 138
38 Target Value Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
39 Estimated Coefficients under Target Settings . . . . . . . . . . . . . . . . . . 141
40 Mean and Standard Deviation of Total Execution Cost (Case-55.187%) . . . 148
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41 Market Impact Parameters (Time Interval: 5 minutes) . . . . . . . . . . . . . 154
42 Market Impact Parameters (Time Interval: 10 minutes) . . . . . . . . . . . . 154
11
Abstract
This research discusses optimal limit order strategy for large institutional investors.
Institutional investors are influential for the market due to their large position. In order
to hold position for relatively long term, the position should be checked and modified
regularly. This modification is called portfolio rebalance. On the other hand, companies
are monitoring actions and motives of institutional investors since their positions could
influence managerial decision frequently. Therefore, strategy of institutional investor is
a key topic for financial market.
In general, there are two types of order. One is market order and the other is limit
order. In case of market order, the investor decides volume. The order will be executed
immediately. However, if the order volume is large, it could change balance between
supply and demand. As a result, market might be changed negatively. This risk is called
market impact. On the other hand, unpredictable price change is also critical risk. If
the execution takes long time, this unpredictable change could make huge cost. This is
called timing risk. These two risks are important for market order strategy. The other
order type, limit order, could mitigate market impact. The limit order decides both price
and order volume. Since this order type gives liquidity to the market, its market impact
is smaller than market impact in many cases. However, if the market price does not
hit the price of the limit order, the order cannot be executed. This uncertainty in order
execution is called unexecution risk. Investors using limit order should care unexecution
risk together with market impact and timing risk.
Despite unexecution risk, using limit order is a good option for large investors to
reduce execution cost caused by market impact further. Through optimizing trade-off
12
relationship among three risks, optimal split of the target volume will be decided. In
some previous literatures about limit order strategies, market impact of limit order is
not considered although it exists. And also, unexecuted limit order cannot be replaced
before maturity. Such previous literatures uses market order at maturity if unexecuted
order volume remains and this could make huge market impact cost. Therefore, this
research considers market impact of limit orders and reorder by limit orders during the
time horizon to capture execution cost more precisely.
Market impact coefficients for limit and market orders are estimated using linear
regression model. This model extends framework of previous literature. In this research,
limit and market orders are separately considered for price change. Through estimating
these coefficients using tick by tick data from the Tokyo Stock Exchange, impacts of
orders are able to be considered correctly. According to the result, it is shown that
market impact of limit order is smaller than that of market order.
Using estimated parameters, optimal strategy is calculated. For implementation, the
Multi-Period Stochastic Programming is used. This Monte-Carlo simulation method
enables optimal strategy to have optimal limit order size in each period. When the market
impact coefficients are sufficiently small, all target volume will be posted from the first
period and all unexecuted volume will be replaced from the second period. By changing
some parameters, sensitivity to optimal solution is also discussed. Under some parameter
settings, target is separated into small pieces and optimal solution posts them separately
in order to reduce some types of risks.
13
(In Italian)
Questa ricerca presenta una strategia ottimale - basata su limit order - per l’acquisto
o la vendita di grandi volumi azionari da parte di investitori istituzionali. L’utilizzo
di limit order può essere una valida opzione per grandi investitori al fine di ridurre il
costo atteso di transazione. Tuttavia, per modellare una strategia ottimale basata su limit
order occorre tener conto (i) dell’impatto negativo che grandi ordini possono avere sul
mercato, (ii) dei tempi di esecuzione potenzialmente lunghi dei limit order che possono
risultare critici, (iii) del rischio di non esecuzione che rappresenta il rischio tipico per
i limit order. Attraverso l’ottimizzazione del trade-off tra questi tre rischi, viene decisa
la suddivisione ottimale del volume target. Gli studi precedenti sulle strategie basate su
limit order hanno in genere trascurato l’impatto sul mercato dei limit order, sebbene in
realtà esso sia tutt’altro che trascurabile. Inoltre, spesso non si considera che la parte
di limit order non eseguita deve essere ritirata e rimpiazzata da un nuovo ordine, in
genere ad un livello di prezzo diverso, più attrattivo per la controparte e quindi meno
conveniente. Pertanto, questa ricerca, che considera l’impatto sul mercato dei limit order
e la necessità di rimpiazzo degli ordini, è in grado di catturare i costi di transazione in
modo più preciso.
I coefficienti di impatto sul mercato dei limit order e dei market order sono stati
stimati utilizzando il modello di regressione lineare. Da questo punto di vista, il nos-
tro modello estende la letteratura precedente ammettendo un diverso impatto sul prezzo
dei limit order e dei market order. La strategia ottimale è stata calcolata utilizzando i
parametri stimati su dati tick by tick della Borsa di Tokyo, prendendo in considerazione
titoli con diverso grado di liquidità.
14
Per l’implementazione, viene utilizzata la programmazione stocastica multi-periodale.
I risultati mostrano che quando i coefficienti di impatto del mercato sono sufficiente-
mente bassi, è conveniente offrire l’intero volume target nel primo periodo, sostituendo
tutti i volumi non eseguiti nel secondo periodo, e così via.
Se l’impatto è maggiore, la strategia ottimale si modifica, in maniera maggiore o
minore a seconda degli altri parametri (tempi di esecuzione dei limit order, rischio di non
esecuzione, impatto differenziale di limit order e market order). Per alcune impostazioni
dei parametri, è conveniente separare il volume target in piccole parti e la soluzione
ottimale consiste nell’offrire tali parti separatamente al fine di ridurre il valore atteso dei
costi di esecuzione.
15
1 Introduction
1.1 Background
In the current society, each of us is required to manage his or her own money. One possible
solution is the asset management. However, only making a deposit in a bank is not sufficient.
Main reason is low interest rate. Interest rate is very low and almost zero or negative in some
countries. For example, according to the European Central Bank [2018], the deposit facility
rate is negative from June in 2014. From March 2016, it is -0.4%. Dynamics of this deposit
facility rate is summarized in Figure 1.
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Figure 1: Dynamics of Deposit Facility Rate in ECB
Due to this low interest rate, it is not enough only to deposit money at banks. This trend
16
is the same in all over the world. Therefore, aggressive asset management is needed.
And also, needs for insurance is increasing because of the aging society. As medical
science develops, many people are able to expect longer lives. On the other hand, the number
of newborn children is decreasing in some developed countries. For example, according to the
cabinet office of Japan, life expectancy is over 20 years longer than that of 1950. Difference
of life expectancy of male and female between 1950 and 2010 is summarized in Figure 2.
Figure 2: Average Life Expectancy of Japan (Source: Cabinet Office of Japan [2017])
By considering these two effects altogether, public support of financial aid might not be
enough. Especially, the amount of pension which is critical for retired people is decreasing.
In order to have sufficient money to live longer, people are required to do asset management.
In order to deal with these problems, institutional investors are necessary. For example,
pension funds, insurance companies, trust banks, and asset management companies are able
to be categorized as institutional investors. Usually, they hold huge amount of assets in order
17
to hold for relatively long term. In case of banks, they are required to invest the money
gathered from retail or wholesale businesses. Insurance companies, on the other hand, are
able to invest insurance fee from huge number of clients. Since their investments are for
their clients, they do not take risks aggressively. And also, they are not required to trade for
short-term in order to deal with longer cash flows.
Due to their huge positions, the institutional investors have responsibility for entire soci-
ety. For example, the Economist [2017] argues that impact investment is required in recent
years. According to this article, impact investing is a type of investment whose concepts are
not only for financial returns but also for social or environmental benefits. If their positions
are so large that they obtain rights of management of the companies, institutional investors
are able to influence the companies. Another article from the Economist [2015] discusses
that management rights in airlines owned by institutional investors could cause environmen-
tal problems.
Some previous literatures also discuss effects of institutional investors to management of
the company or entire society. For example, Gillan-Starks [2000] argues that judges by insti-
tutional investors are more influential than individual investors. On one hand, such influences
are able to improve performance of the company positively. Basically, institutional investors
are assumed to oppose antitakeover of the company (Brickley et al. [1988]). Hartwell-Starks
[2003] concludes that institutional investors are able to mitigate agency problem between
shareholders and managers. What is more, positive correlation exists between percentage of
stock holdings by institutional investors and shareholder activism, governance structure, and
operating performance (Smith [1996]).
18
On the other hand, each company behaves so that institutional investors decide to buy
the assets of the company. Dhaliwal et al. [2011] shows that institutional investors are more
interested in the companies who disclose CSR reports. Other literature by Ajinkya et al.
[2005] concludes that companies which have more institutional investors or outside directors
report performance of the company more precisely, accurately, and frequently.
Due to their influential position in our society, many institutional investors set investment
policies. For example, the Government Pension Investment Fund [2015] who is one of the
largest pension funds in the world declares that they try to fulfill their steward ship respon-
sibilities such as ESG (Environment, Social, and Governance) investments. They also show
clearly that their medium- to long-term returns are for pension recipients.
In stock exchange market, there are two types of investors. One does trading and the
other does execution strategy. Above mentioned institutional investor basically uses execu-
tion strategy. In case of trading, traders both buy and sell assets. Through buying low and
selling high, they are able to earn money which is the difference of buy and sell prices. Typ-
ical type of trading is the High Frequency Trading (HFT). HFT buys and sells single asset
very frequently. Such traders often post orders hundreds and thousands of times in a second
thanks to the high-speed Internet access, software, and hardware developments. They care
inventory level and try not to hold large position. If their inventory level becomes large, the
inventory could be influenced by unpredictable fluctuation of market price. In case of posi-
tive inventory, value of the inventory could be decreased if the market price declines. If the
level is negative, the negative inventory could force traders to buy the asset at higher price. In
the end of the trading period, inventory level should be managed to be zero so that the trader
19
does not have to care market price.
On the other hand, investors using execution strategy try to trade fixed amount of asset in
a fixed time horizon, such as in one hour or in a day. The target amount is already decided
through considering their position and market condition at the time. They either sell or buy
the asset. What is more, position of such investors is often huge and their transactions are
also large size which are costly for them. Therefore, main objective of such investors is to
minimize execution cost through deciding timing and order volume in each period. Charac-
teristics of these two types of traders are summarized in Table 1.
Table 1: Main Types of Traders
Trading Execution Strategy
Objectives Return Transact Fixed Amount
Direction of Transaction Both Buy and Sell Either Buy or Sell
Target Maximize Return Minimize Execution Cost
Position Small Large
Typical Player High Frequency Trader Institutional Investor
20
1.2 Objective of this Research
In this research, execution strategy for institutional investors is discussed. By introducing
some new concepts for execution model, execution cost is tried to be understood more pre-
cisely and be reduced. Through reducing cost, institutional investors are able to invest more
efficiently.
1.3 Structure
Before discussing the model, some basic concepts will be explained in section 2. Here, order
types and risks are introduced.
In section 3, some previous literatures and optimization model are introduced. Firs, some
literatures discussing optimization models are explained. Among many researches, a few
literatures will be explained deeper. After that, novelties are listed in section 3.6. Section 3.7
describes framework of the model.
Among variables used in the model, some parameters are needed to be estimated and this
will be shown in section 4. In section 4.1, some previous literatures discussing parameter
estimations are introduced. Section 4.2 explains objectives and methodologies of parameter
estimation through considering optimization model and condition of market data. After that,
result of parameter estimation will be illustrated in section 4.3.
Through implementing estimated parameters, section 5 introduces result of optimization.
By changing some parameter settings, sensitivity of the optimal strategy will be discussed in
section 5.1, 5.2, and 5.3. Finally, all parts are concluded in section 6.
21
2 Basic Concepts
In this section, some basic concepts about execution strategy are introduced. First, market
order and limit order are explained. Then, three risks in order to consider execution strategy
are illustrated. These risks are market impact, timing risk, and unexecution risk.
2.1 Market Order and Limit Order
There are two types of orders, one is market order and the other is limit order. Market order
decides volume to buy or sell. It cannot order price to trade. In case the best bid price is
100 yen and the best ask price is 101 yen, trader is able to buy assets at 101 yen if using
market buy order. In case of sell order, the assets are sold at 100 yen. Market order will
consume pooled limit orders in the opposite side of the book. Although the market order will
be executed immediately, market order requires cost.
On the other hand, limit order decides both volume and price. For example, if the investor
posts limit buy order at 100 yen, the posted order will be executed only if the best price goes
below 100 yen. If the best bid price goes above 100 yen or the orders posted before the
investor’s order are not executed. In case of limit order, price is decided and there is no
uncertainty in price. However, risk of not being filled should be considered.
2.2 Market Impact
First, market impact is a specific risk for huge institutional investor that his orders could affect
market negatively. In order to modify his position, institutional investors might buy or sell
22
large volume of stocks. Such large orders could disturb balance between supply and demand.
If large buy order comes to the market and demand becomes larger than supply, sell orders
which are pooled in the book are consumed. Consequently, demand exceeds supply and price
could increase. This change forces buy investors to trade at unexpectedly and not preferably
higher price. This is called market impact. Through this market impact, execution cost for
institutional investor could increase. Although single small order makes almost no market
impact, cumulative execution cost will increase if total target order volume becomes huge.
Many institutional investors try to make execution strategy in order to avoid this impact and
not to bear unexpectedly high cost.
2.2.1 Market Impact of Market Order
For example, if the investor would like to post 1000 volume of market buy order, this will be
executed immediately. If the best ask price is 200 yen and its depth is 3000 volumes, 1000
volume among 3000 volumes of the depth is consumed. Consequently, he is able to buy 1000
stocks at 200 yen. This transaction is illustrated in Figure 3.
23
Figure 3: Market Order without Market Impact
If the market order volume is larger than the best price depth, all limit orders in the best
price is consumed and the best price will be changed. For example, if there are only 500
stocks at the best ask price, 200 yen, not all 1000 stocks are able to be executed at 200 yen.
In this case, only 500 stocks can be bought at 200 yen. After that, another 500 stocks will be
matched with the second best ask price, for example, at 201 yen. If the depth of 201 yen is
over 500 stocks, they can be executed at 201 yen. To sum up, 500 stocks are bought at 200
yen and another 500 stocks are bought at 201 yen. In this case, the best price is changed to
be 201 yen. This is called market impact. Therefore, market order could have large market
impact if the order volume is relatively large. Image of the market order when market impact
exists is shown in Figure 4.
24
Figure 4: Market Order with Market Impact
2.2.2 Market Impact of Limit Order
If the best bid depth at 199 yen is 2000 and the limit buy order is 1000, the best bid depth
will increase to 3000. Image of this limit order to 199 yen is described in Figure 5.
Figure 5: Limit Order
25
In case of limit order, it does not have market impact if the order is executed at the price set
by the investor. However, if the limit order cannot be executed, he is forced to post the order
to less favorable price in order to hold higher execution priority than the previous limit order.
Even in this case, market impact is smaller than the market order because the limit order does
not consume liquidity and it does not have to cross the bid and ask spread. Therefore, using
limit orders could allow investors to trade with lower execution cost.
Limit order does not have market impact if market order volume coming from the opposite
side of the limit order book is large. In this case, the limit order could be filled immediately
and completely. And also, there are no price impact and unexecution risk. However, if the
arriving market order volume is not sufficient, unexecution risk should be considered. In this
research, investor is assumed to cancel his order and replace it to the new best price as a new
limit order in order to hold good execution probability. This strategy is assumed to be the
cause of market impact of limit order.
26
Figure 6: Patterns of Market Impact of Limit Order (LMI)
Using Figure 6, concept of limit order’s market impact is explained. Here, 3000 units
of limit buy order are posted to price 100. This price is assumed to be the best price. In
case there are 1000 units of market order coming in, 1000 limit orders can be matched and
they are executed at price 100. However, remaining 2000 limit orders are not filled at price
100. These will be cancelled and placed to price 110 which is worse than price 100 for the
investor. However, price 110 is better for the traders of the other side and the posted order
is assumed to have higher execution probability than price 100. If another 1000 units of
market order exist, 1000 limit orders out of 2000 are executed and other 1000 units are still
unexecuted. Therefore, the investor will cancel these 1000 orders and post them again to
price 120. Finally, the last 1000 orders are assumed to be executed at price 120. In this case,
even using limit orders, price changes from 100 to 120 and this change is defined as market
impact of limit order. Execution probability of this case is assumed to be 100 %. If the final
27
1000 units of order at price 120 are not executed, 2000 orders are executed in total and its
execution probability is 66.7%. In case no market order comes, as it is shown in the right
figure, price is not changed and market impact will not occur. Execution probability is zero
in this case. In order to model this example, we assume linear price impact to executed order
volume. In the later section, this assumption is modified that price impact is assumed to be
linear to posted order volume.
2.3 Timing Risk
Then, not only market impact, but also timing risk should be considered in order to develop
execution strategy. In general, market impact is able to be reduced through splitting the large
order volume into small pieces. However, posting many small orders requires long time until
achieving the target. If completion of the execution requires long time, remaining unexecuted
order volume could be influenced by unpredictable price fluctuation. The unpredictable price
change is able to be characterized by volatility of fundamental stock price. This risk is called
timing risk.
2.4 Unexecution Risk
The final risk, unexecution risk, occurs only to limit order. The stock exchange market is
organized in response to price and time priority. If an investor posts a buy (sell) limit order,
the other buy (sell) orders posted to higher (lower) price or prior to the investor’s order will be
prioritized and be matched first. The probability of finishing the trade without being executed
28
could increase if the limit order size being pooled is already large. And also, if the amount of
incoming market order volume is small, the limit order may not be executed before maturity.
This risk is called unexecution risk.
One literature by Foucault [1999] explains this risk. According to this research, limit
order could trade at better price. However, using limit order has risk in unexecution and
winner’s curse problem. Winner’s curse problem is a typical risk of auction. When many
players bid, a winner could be forced to pay more than expected since all players do not
know market value.
In conclusion, using limit orders is a beneficial strategy for huge institutional investors.
However, limit order has market impact, timing risk, and unexecution risk. By considering
trade-off relationship among these three risks, optimal execution strategy is constructed.
2.5 Three Risks and Order Types
By summarizing order types and their risks, following table 2 can be drawn.
29
Table 2: Order Types and Their Risks
Market Order Limit Order
Decision ONLY Size Price and Size
Advantage Immediate Execution Lower Cost
Disadvantage Higher Cost Unexecution Risk
Market Impact *** *
Timing Risk * *
Unexecution Risk - ***
(***: Critical, *: Important, -: Not Important)
30
3 Optimal Execution Strategy Model
3.1 The Limit Order Book and Two Order Types
Some previous and traditional literatures discussing condition of limit order book and strate-
gic choice by investors exist. One previous literature by Biais et al. [1995] shows this trade-
off relationships among three risks. They analyses tick by tick data of Paris stock exchange.
According to this literature, traders decide their strategy based on the condition of the limit
order book. For example, when the depth is thin, the number of limit order increases. When
it is thick, on the other hand, inflow of market order increases. This is because of the unex-
ecution risk. Large depth means that posted limit order is forced to wait long queue to be
executed. In terms of bid ask spread, limit orders will be placed inside the spread when it is
wide.
Parlour [1998] also shows that choices between limit order and market order will be made
according to the market condition. And also, this research concludes that order submission
could influence later order inflow. This influence could result in changes in execution proba-
bility and this is thought to be market impact risk.
Although two types of orders have advantages and disadvantages, one research by Harris-
Hasbrouck [1996] argues that limit order is used more than market order since its perfor-
mance is better through real data analysis of Super DOT, NYSE. This hypothesis holds even
if penalty of unexecuted order is considered. Another literature by Handa-Schwartz [1996]
also supports this result.
Foucault et al. [2005], on the other hand, discuss difference between limit order and mar-
31
ket order from concept of time. Limit order has disadvantage in order execution. It usually
takes longer time to be executed than market order. Therefore, this literature concludes that
patient trader chooses limit order but the other not patient trader chooses market order.
In any case, optimal strategy should be constructed through considering advantages and
disadvantages or trade-off relationship among risks.
3.2 Literatures about Limit Order Book
In this section, some previous literatures discussing limit order book are introduced. The
literature by Cho-Nelling [2000] analyses tick by tick data of NYSE. According to this litera-
ture, execution probability of limit orders depends on market conditions. For example, when
posting sell orders, posting order far away from the best price, posting large orders, or being
in the low price volatility situations, execution probability of limit order is low. They also
show that probability of limit orders to be filled is not increased even if the trader holds the
limit order for a long period.
Some other literatures discuss shape of limit order book. Bouchaud et al. [2002] show that
limit order book is power-law shape through analysis of Paris stock exchange data. Another
literature by Ranaldo [2004] argues that shape of the limit order book could affect strategy
of traders. When their own side (other side) book is thick (thin), or temporary volatility is
high, trading becomes aggressive. Under aggressive order inflow condition, spread becomes
smaller and market impact risk becomes smaller as well (Rou [2009]).
In the later extensions, models of limit order book are discussed. For example, Cont-
De Larrard [2013] make order inflow model using the Markovian Queueing System. Their
32
model is able to discuss interval between price changes, autocorrelation of price change, and
probability of upward price jump. Muni Toke [2015] and Frino et al. [2017] relate limit order
book models to execution strategy model.
3.3 Literatures about Market Order Strategy
In this section, previous literatures of market order strategies are introduced. Mainly, this
topic could be subdivided into two research areas. One is market impact model and the other
is limit order book model.
Berstimas-Lo [1998] is one of the most famous literatures of market impact model. This
literature will be introduced more precisely later. Moazeni et al. [2013] is another literature
of market impact model. This considers jump-diffusion process in price dynamics. Through
considering this concept, this research minimizes sum of the expected execution cost and its
conditional Value at Risk (CVaR). CVaR will be explained later.
If market impact function is investigated more, Malik-Ng [2014] estimate this function
from limit order book information. They conclude that market impact function is nonlinear,
time varying, and asymmetric. Market impact function could be influenced by order imbal-
ance according to Easley et al. [2015]. In terms of seasonality within a day, Wilinski et al.
[2015] argue that market impact is large in the beginning of the day and small in the end of
the day. This result is obtained through analysis of LSE data and immediate price impact
function analysis.
According to these market impact function models, many literatures such as Moreau et
al. [2017] propose execution strategy model. Some of them will be introduced in the next
33
sections.
Berstimas-Lo [1998] (Market Order Strategy)
Many previous researches discussing optimal execution strategy for institutional investors are
mainly focused on market orders. Berstimas-Lo [1998] optimize market order strategy.
In this literature, target asset volumey is separated into small pieces in order to trade
within given number of intervalN using market orders. The market orders are assumed to
have market impact. Price dynamics is defined as follows. Stock price at period t is shown as
Pt . Price impact coefficient is illustrated asθ which is linear to order volume at each period
zt .
Pt = Pt−1 + θzt + σξt (1)
Price dynamics has two components. One is price impact and the other is fundamental
process. The fundamental process is characterized as white noise withσ standard deviation.
Using this price dynamics definition, optimization model is defined as follows.
min. E1
N∑t=1
Pt · zt (2)
s.t.∑
zt = y (3)
0 ≤ zt (4)
Pt = Pt−1 + θzt + σξt (5)
E[ξt |zt, Pt−1] = 0 (6)
In this literature, the number of shares to be purchased which is defined asyt is used to
34
optimize the model.yt is described as follows.
yt = yt−1 − zt (7)
y1 = y (8)
yN+1 = 0 (9)
Using these notations, Bellman equation is implemented.
Vt (Pt−1, yt ) = minzt
Et[Pt · zt + Vt+1(Pt, yt+1)] (10)
By minimizing this value function recursively, optimal order size and value function is
calculated as follows.
z∗1 =y1
N=
y
N(11)
V∗1 (P0, y1) = y1(P0 +N + 12Nθy1) (12)
= P0X +θy2
2(1+
1N
) (13)
This result shows that optimal order size in each period is the same. The volume is
decided by splitting the total target volumey into equal size.
From this point, the literature extends the model to include linear price impact with infor-
mation defined as follows.
Pt = Pt−1 + θzt + oMt + σξt (14)
Mt = ρMt−1 + ηt (15)
The exogenous parameterMt is assumed to be information which influences the stock
price. For example, return of S&P 500 index can be used as this factor. By including this
information, the solution changes a bit.
35
What is more, multi stock investment is also discussed. Usually, institutional investors
assumed to execute some assets in order to modify their position. In some cases, stock prices
have correlation. Therefore, correlation could influence the strategy negatively and positively
according to the characteristics of the assets.
Although this model is a basic starting point, there are some points that can be extended.
For example, objective function is simple expected execution cost. However, this index is not
suitable to capture tail of the cost distribution. Therefore, risk measure such as variance or
CVaR can be implemented. And also, this model uses market order for execution. In order to
consider limit order, execution probability should be included.
Almgren-Chriss [2000] (Market Order Strategy)
The next literature by Almgren-Chriss [2000] discusses also market order strategy. This re-
search introduces new concepts of market impact which are temporary and permanent market
impact. As in the literature by Berstimas-Lo [1998], target volume is given byy units and
this is required to be executed within given maturityT. The number of trading periods is
given byN. Trader posts fractions of target order volume everyt = TN interval.
Price dynamics of the asset evolves according to volatility and drift which are exogenous
factors, and market impact which is endogenous factor. It is defined as follows. Remaining
to be executed order volume in periodk is shown asyk and posted order volume is defined
36
aszk.
Pk = Pk−1 + σ√
tξk − τg(zk
t) (16)
P̃k = Pk−1 − h(zk
t) (17)
Posted order volume can be written aszk = yk−1 − yk. In terms of exogenous factors
ξ, they are given independently and randomly from the behavior of the target institutional
investor.
The market impact has two components, one is temporal and the other is permanent. The
term g(·) shows permanent market impact. The other termh(·) shows temporary market
impact.
Temporary market impact is temporal change of supply and demand balance caused by
the institutional investor, whereas permanent impact remains at least during the entire trading
period.
Its objective function is sum of the expected shortfall and the variance of shortfall multi-
plied by the risk aversion coefficient.
Obj = E[x] + γ · V[ x] (18)
=∑
tykg(zk
t) +∑
zkh(zk
t) + γσ2
∑ty2
k (19)
By solving constrained optimization through introducing the Lagrange multiplier, strat-
egy is optimized. If the risk aversion coefficient is zero, only expected shortfall is considered
and optimal strategy becomes equal size execution. On the other hand, if the risk aversion
coefficient is positive and variance of the shortfall is also considered, large volume of order is
37
posted in the first period. Through reducing remaining order volume aggressively, variance
of shortfall is able to be reduced.
The framework of considering both expectation and variance of execution cost is assumed
to be better than only considering expectation as in the previous section. Therefore, these
terms can be used also in the limit order book model.
3.4 Literatures about Limit Order Strategy
Although the number of literatures discussing limit order strategy is not large, some litera-
tures exist. One of the most famous and traditional research of limit order strategy is Esser-
Monch [2007]. This will be introduced later. Another research by Lo et al. [2002] makes
limit order model including survival analysis and real data analysis. The strategy is able to
be characterized by price, bid ask spread, order size, and volatility. Although they model
these four parameters, they conclude that price could influence more than order size to opti-
mal strategy. Another literature by Gueant et al. [2012] make execution model which could
decide price range as well. They insist that the model is able to consider unexecution risk
and price risk. Nystroem et al. [2014] use almost the same concept but adding uncertainty
in limit order book. Finally, Frey-Sandas [2017] show that the Iceberg strategy which is a
typical strategy using limit orders could influence liquidity positively. They also show that
unpredictability in limit order book could negatively affect the strategy through adverse selec-
tion cost. Although this literature uses different concept, this result infers that market impact
of limit orders is assumed to exist and needed to be considered carefully.
38
Esser-Monch [2007] (Limit Order Strategy)
As we have surveyed so far, research of optimal limit order strategy for institutional investor
is not popular compared with market order strategy. One common research focusing on
institutional investor is done by Esser-Monch [2007]. Focus of this research is the Iceberg
strategy. Iceberg order is a limit order strategy for large institutional investors who try not
to show that they are large investors and would like to execute large volume. Showing that
they are large could sometimes induce large execution cost if other traders try to exploit this
opportunity. Therefore, institutional traders are required to hide their characteristics through
dividing large target order volume into small pieces.
In this literature, waiting time to be executed and adverse informational impact caused by
showing large limit order are assumed to be trade-off relationship. Stock price is modelled as
geometric Brownian motion.
dPt = µPtdt + σPtdWt (20)
In this model, price impact is assumed to be constant and independent of order size posted
by the institutional investor. Price will be jumped if the best price hits the price where iceberg
order is placed. Size of the jump is modelled asϵ . After hitting the price, price will be
changed as follows.
Pt− = P̄ (21)
Pt = (1− ϵ )P̄ (22)
Through minimizing execution cost under iceberg strategy settings, the research discusses
optimal limit order strategy. Although this notation is a simple modelling for market impact
39
of limit order, this assumption is not realistic. In the optimization model introduced later,
market impact is assumed to be linear to the executed order volume or posted order volume.
The posted iceberg limit order will be executed if and only if the price hits the limit price.
Here, target order volume isϕ0, peak of the limit order is shown asϕp, depth of the limit
price which is assumed to be flat in all price range isϕa, and the each executed order volume
when the limit price is hit is described asϕs. In this research,ϕs amount will be executed
every time when the best price hits the target price. By introducing these variables, sufficient
number of hits of the pricen∗ can be written as follows.
n∗ = ⌈((ϕ0/ϕp)ϕa + ϕ0)/ϕs⌉ (23)
The objective function is defined as execution cost.
max ϕ0P̄ (24)
s.t. P∗ ≤ P[n∗ ≤ M] (25)
P0 < P̄ (26)
ϕp ≤ ϕ0 (27)
By solving this problem numerically, optimal answer is obtained. When implementing
parameters estimated from real market data, the optimal solution is divided into two parts
with different order sizes. In the beginning of the day, order volume is large. In the last half
of the day, order volume becomes a half.
In this model, however, cannot capture partial execution, market impact of limit order,
and price dynamics precisely. For example, the same amount will be executed every time
40
when the best price hits the target. This assumption is not realistic and should be discussed.
And also, this model is not clear about the depth. The limit order book is assumed to be flat in
all price range. However, its depth is renewed every time with the same size before the order
submission of institutional investor. This point should also be discussed. In the proposing
model, execution probability is defined and used to capture order flow dynamics.
Agliardi-Gencay [2017] (Limit Order Strategy)
More recently, Agliardi-Gencay [2017] propose an optimal limit order strategy model con-
sidering both price distance from the best price and size of each order. Considering price
distance from the best price is a typical framework for the High Frequency Trading strategy.
In this model, the target order volumey will be executed within N number of periods. In
each period,zk will be placed as limit order. The limit price of period k is defined asP̂tn is
written as follows.
P̂tn = Ptn + δn (28)
δ shows price distance from the best price at the momentPtn . Based on the order size and
price distance, execution probabilityΛ is defined as follows.
Λ(z, δ) = A · exp(−kz− hδ) (29)
k andh are parameters for execution probability. Distribution of execution probability is
modelled as exponential function. By using exponential function given order size and dis-
tance from the best price, execution probability is able to be modelled where the probability
is small if the posted order volume is large and price distance is far from the best price.
41
When the order size is large and the distance is far away from the best price, probability to
be filled completely decreases exponentially. Using this distribution of execution probability,
remaining unexecuted order volume can be defined as follows.
yk = y −k∑
j=0
I j zj (30)
I j is a parameter for execution. If it is one, the orderzj is fully executed. If it is zero, the
order is assumed not to be executed. Terminal wealth which can be obtained through limit
orders is defined as follows.
RN = yPN +
N−1∑k=0
zk Ik(δk + Pk − PN) (31)
Expected terminal wealth is written as in the following formula.
E0[RN] = yP0 + E0[N−1∑k=0
zkδkΛk] (32)
This model, however, did not consider the price impact since each order volume is small
and the small orders are assumed not to influence entire market. And also, this research
assumes complete execution. This means that the posted order is executed completely or not.
If some order cannot be executed within maturity, the remaining unexecuted order volume
is assumed to be placed as market orders. The penalty of using market orderl is used and
cost for using market order can be written asl2y
2N−1
Objective function is sum of the expected execution cost and the risk of stock price fluc-
42
tuation. The risk and objective function are defined as follows.
N−1∑k=0
yk(Pk+1 − Pk) =N−1∑k=0
yk∆Pk (33)
Risk = E0[N−1∑k=0
Varn(yk∆Sk)] (34)
Obj = max E0[N−1∑k=0
(zkδkΛk −γ
2Vark[yk∆pk]) − l
2y2
N−1] (35)
Through backward induction, optimal strategy ofzk andδk are calculated.
Through numerical implementation of the optimal model, this literature illustrates effects
of some parameters to the optimal strategy.
We do not consider price distance from the best price since price range is discrete and
this framework forces the distance to be continuous. And also, market impact of limit order
is not considered although it exists. What is more, all unexecuted order volume will be
replaced as market order at maturity which is too costly. Therefore, we model reorder strategy
which allows investors to replace unexecuted limit order from the next periods as limit orders.
Through aggressive reduction of unexecuted order volume through limit order, this model is
able to reduce execution cost.
Hautsch-Huang [2012] (Market Impact of Limit Order)
The literature by Hautsch-Huang [2012] discusses market impact of limit orders. All previous
literatures of limit order strategies introduced above do not consider market impact of limit
orders. However, Hautsch-Huang [2012] conclude that not only market orders but also limit
orders could have price impact based on market data analysis. This price impact could be one
of the most important costs especially for large institutional investors.
43
Hosaka [2014] (Existence of Cancel Order)
In terms of trading strategy, cancel orders could be considered for optimization. According
to Hosaka [2014], over 40% of orders in the Tokyo Stock Exchange are cancel orders. This
order type cancels limit orders which the trader have previously posted to the market, not
been executed, and been remaining in the market. This cancel order is assumed to be aimed
at cancelling limit order once and posting it again to the price which has higher execution
probability in order to execute the limit orders quickly.
3.5 Literatures about Limit Order Book Strategy
Before introducing novelties, some literatures of limit order book strategy are introduced.
These literatures use another aspect of market impact, limit order book dynamics. Through
considering reaction after posting an order, optimal strategy will be constructed. Literature
by Farmer et al. [2004] shows that existence of some prices without any orders could result
in huge market impact if the free price is crossed. Obizhaeva-Wang [2013] is one of the most
famous and traditional literature of limit order book strategy model. This shows that static
characteristics such as the bid ask spread and depth are not influential to optimal strategy.
Dynamic characteristics such as resilience, on the other hand, influences more than static as-
pects. The research also shows that large order consumes liquidity already exists but induces
new order inflow, whereas small order is absorbed by incoming liquidity.
Based on this framework, Bayraktar-Ludkovski [2014] make a model where limit or-
der book decays exponentially as going deeper in the book. Lin-Fahim [2017] extends
44
Obizhaeva-Wang [2013] so that inflowing liquidity is assumed to be finite which is more
realistic than previous framework.
This limit order book model will not be considered in the proposing model. However, by
discussing this model more, some problems of this literature could be solved. In any case,
these will be future extensions and discussed later.
3.6 Novelties of this Research
In this research, price impact of limit orders is assumed to exist. If the institutional investor
would like to post large order volume which is critical for balance between supply and de-
mand, it could have market impact. In order to consider this assumption, optimal strategy is
constructed using the Multi-Period Stochastic Programming Model (MPSP). And also, can-
cel order is considered. During the trading period, unexecuted orders can be cancelled once
and posted again later in order to reduce unexecution risk. Also, market impact and other
relevant parameters are estimated using tick by tick data of Tokyo Stock Exchange. By using
estimated parameters, this paper is able to discuss real application of the execution model.
The main novelties are these three points.
1. Compute optimal limit order strategy involving unexecution risk.
The unexecuted order will be executed at maturity by market order. However, the
market impact of market order is greater than limit order and investors might be suffered
from it if all unexecuted orders are executed using market order. In this research,
limit orders are preferred and used as much as possible using reorder and the CVaR
45
(Conditional Value at Risk) as a risk measurement.
2. Market impact of limit order is considered
The previous researches did not involve price impact of limit orders although it exists.
In this research, limit order is assumed to have negative impact to the price.
3. Consider reorder for unexecution risk.
In this research, replacement of limit order is considered. Even if the posted limit
order cannot be executed, it can be cancelled and can be replaced to new price which
has higher execution probability. Through this reorder, institutional investors are able
to reduce unexecution risk aggressively and terminal market order volume becomes
smaller.
4. Sensitivity of optimal strategy by various market conditions are evaluated
Since liquidity is different in each stock, execution probability and market impact are
also different. In this research, sensitivity analyses of various parameters are done
and they clear out how optimal strategy will be changed through changes of market
condition.
5. Illustrate practical application of the execution model through parameter estimation.
Some parameters such as market impact and execution probability are estimated using
market data. Through implementing real parameters, this research shows how to use
the execution model in some different assets.
Comparison of this research with previous literatures is summarized in Table 3.
46
Table 3: Comparison with Previous Researches
Esser-Monch [2007] Agliardi-Gencay [2017] This Paper
Approach DE* DE* MPSP**
Solution Analytical Analytical Numerical
Objective Function Min. Expected Cost Max. Net Wealth Min. E[Cost]+CVaR
Price Continuous Continuous Discrete
Price Impact Constant No Linear
Reorder No No Yes
(DE*: Differential Equation, MPSP**: Multi-Period Stochastic Programming)
47
3.7 Objectives, Research Methodology and Research Framework
In this research, main target is an institutional investor who would like to tradey units of
single stock before terminal period t=N. In general, the investor either buys or sellsy units or
stock. Here, the investor tries to buyy units of stock in the given trading period. The model
decides optimal limit buy order size in each periodt = k(k = 1, · · · , N). In this research,
each limit order is assumed to have market impact. This market impact is assumed to be
linear to the executed order volume. If some unexecuted order volume exists afterN number
of limit buy orders, they will be executed by market order at maturity. Through this strategy,
total traded volume could achievey units completely. The execution cost is defined using
difference between actual executed price and initial price. The objective function is defined
as sum of the expected execution cost E[CN+1] and CVaR of total execution cost multiplied
by the risk aversion coefficientγ.
3.7.1 Model
Parameters and formulas of this model are illustrated as follows.
Indices
• I : Total Number of Simulation Paths
• N: Total Number of Periods
• i : Path Number (i = 1, · · · , I )
• k: Discrete Period Number (k = 1, · · · , N + 1)
48
Random Numbers
• ξ (i )k : Random Number from Standard Normal Distribution of i-th Path, k-th Period
• τ(i )k : Percentage of Executed Order of i-th Path, k-th Period
Parameters
• p0: Initial Price of the Stock
• MMI : Market Impact of Market Order at Maturity
• LMI : Market Impact of Limit Order
• β: Confidence Interval for CVaR
• γ: Risk Aversion Coefficient
• y: Target Stock Volume
Decision Variables
• xk: Total Posted Order Volume of k-th Period
Intermediate Variables
• aβ: VaR at Confidence Intervalβ
• u(i ): Expected Loss of i-th Path
• w(i )k : Executed Order Volume of i-th Path, k-th Period
49
• y(i )k : Total Amount of Orders to be Executed of i-th Path, k-th Period
• z(i )k : The Actual Posted Orders of i-th Path, k-th Period
• P(i )k : Stock Price of i-th Path, k-th Period
• C(i )k : Cumulative Execution Cost of i-th Path, k-th Period
Overall Model
minx1,··· ,xN
E[CN+1] + γ · CVaR[CN+1] (36)
s.t . w(i )k = τ
(i )k z(i )
k (37)
z(i )k = min(y(i )
k−1, xk) (38)
z(i )N+1 = w(i )
N+1 = y(i )N (39)
y(i )k = y(i )
k−1 − w(i )k (k = 1, · · · , N + 1) (40)
P(i )k = P(i )
k−1 + P(i )k−1 · (LMI · w(i )
k + σξ(i )k ) (k = 1, · · · , N) (41)
P(i )N+1 = P(i )
N + P(i )N · (MMI · w(i )
N+1 + σξ(i )N+1) (42)
C(i )k = w(i )
k (P(i )k − P0) +C(i )
k−1 (k = 1, · · · , N + 1) (43)
CVaR[CN+1] = aβ +1
(1− β)I
I∑i=1
u(i ) (44)
aβ −C(i )N+1 + u(i ) ≤ 0 (45)
0 ≤ u(i ) (46)
N+1∑k=1
w(i )k = y (47)
y(i )0 = y, y(i )
N+1 = 0, P(i )0 = P0, C(i )
0 = 0 (48)
50
3.7.2 The Simulation Multi-Period Stochastic Programming
This research uses the Simulation Multi-Period Stochastic Programming model which is able
to describe uncertainty and optimize execution strategy through Monte Carlo Simulation (Hi-
biki[2000], Hibiki [2001], Hibiki-Komoribayashi [2006]). In the Simulation Multi-Period
Stochastic Programming model, single optimal solution will be calculated given large num-
ber of simulation paths. In the model settings, N number of optimal limit order volume will
be obtained without considering how much the trader could execute before. Figure 7 shows
image of this stochastic programming model.
Figure 7: Simulation Multi-Period Stochastic Programming Model
Each line and node shows generated simulation path. Horizontal direction shows time. In
each period, all simulation paths are gathered and single optimal order size will be calculated.
This model was introduced by Hibiki [2006]. Multi-Period model is able to set strategy in
some periods. Oppositely, single period model decides only strategy of the first period. And
51
also, the multi-period model is able to describe uncertainty over some periods. Considering
both strategy and uncertainty in some periods, optimal strategy will be constructed. There-
fore, the strategy includes future conditions and estimations. The solution, however, does not
insist the investor to follow it under any conditions. For example, if distribution of execution
probability or unpredictable price change differ from initial estimation very much, optimal
solution should be calculated again later. In the real situation, the strategy might be calcu-
lated some times in order to fit each condition. For future extentions, multi-strategy could be
given to simulated paths. For example, some simulation paths which have almost the same
execution cost could be bundled. During the optimization, different strategies could be given
to those generated nodes. By giving various choices, execution cost could be reduced further.
This point might be discussed in the future works.
Although the strategy will be modified later, multi-period model is different from single
period model. The multi-period model decides strategy in the first period given strategies in
the later periods. This solution in the first period might be different from the strategy obtained
from single period model. Therefore, multi-period model should be used in order to consider
uncertainty over some periods more cleary and strategies in the later periods.
52
3.7.3 Objective Function
The objective function is defined as sum of the expected execution cost and its Conditional
Value at Risk (CVaR) multiplied by the risk aversion coefficientγ. Limit order has unexecu-
tion risk and unexecuted orders could influence distribution of final execution cost. In other
words, tail of the cost distribution could become fat. Therefore, this model implements CVaR
as a measurement of downside risk1. CVaR is able to capture distribution of cost in the larger
side. If using VaR, cost above the VaR cannot be considered.
minx1,··· ,xN
E[CN+1] + γ · CVaR[CN+1] (49)
Downside risk measure CVaR is an average of excess total execution cost from VaR.
This downside measure is able to be calculated using following equations.aβ is Value at
Risk (VaR) which is calculated using confidence intervalβ. u(i ) shows excess amount of
cumulative cost which exceeds VaRaβ.
CVaR[CN+1] = aβ +1
(1− β)I
I∑i=1
u(i ) (50)
aβ −C(i )N+1 + u(i ) ≤ 0 (51)
0 ≤ u(i ) (52)
1Many researches of market order strategy such as Almgren-Chris [2000] use variance as the risk measure-
ment. However, if variance is used for unexecution risk, optimal trading strategy tends to reduce variability
of execution cost. However, lower execution cost should not be considered as risk. Therefore, they post large
orders in the later periods if risk aversion coefficient is not small.
53
Figure 8: Image of Cumulative Cost Distribution and CVaR
By using sample dataset, CVaR can be computed as in Table 4.
Table 4: Example of CVaR Calculation
i 1 2 3 4 5 6 7 8 9 10
C(i )N+1 100 300 500 600 650 700 720 730 750 800
aβ −C(i )N+1 630 430 230 130 80 30 10 0 -20 -70
u(i ) 0 0 0 0 0 0 0 0 20 70
Over VaR F F F F F F F F T T
(VaR=730, The Eighth Smallest Cost)
In this example,β is set to be 0.8. This means that the simulation paths which have
cumulative costC(i )N+1 larger than the eighth smallest cost are considered in the CVaR. In the
example, VaR of 80% is 730. Therefore, the ninth and tenth smallest cost paths, 750 and 800,
are considered. CVaR is the average of these two costs. Therefore, CVaR is 775.
This can also be computed using excessive costu(i ). The table 4 shows thatu(i ) is 20 and
70 in the ninth and tenth smallest cost paths. Therefore,1(1−β)I
∑u(i ) = 20+70
(1−0.8)·10 = 45 and
54
aβ + 1(1−β)I
∑Ii=1 u(i ) = 730+ 45= 775.
3.7.4 Constraints of Executed Order Volume
The posted order volumez(i )k of k-th period is decided through comparing optimized order
volume (xk) and total unexecuted order volume of (k-1)-th period (yk−1). The optimized
order volumexk is not path dependent and will be given by the Multi-Period Stochastic Pro-
gramming model. On the other hand, unexecuted order volume is path dependent given the
execution probabilities in the previous periods. The smaller of these two volumes will be
chosen. If unexecuted order volume (yk−1) is lower than the optimized order volume (xk),
unexecuted order volume is assumed to be small enough and all unexecuted order volume
will be placed. On the other hand, if the remaining unexecuted order volume is too large,
the optimized order volumexk will be chosen and posted since posting all unexecuted order
volume could induce huge market impact. Execution probability is given asτ(i )k . If unexe-
cuted order volume at maturityy(i )N is positive, those will be placed as market order defined
asw(i )N+1.
w(i )k = τ(i )
k z(i )k (53)
z(i )k = min(y(i )
k−1, xk) (54)
z(i )N+1 = w(i )
N+1 = y(i )N (55)
If the posted limit order is not filled, it will be cancelled before the next period and new
limit order will be posted to the new best price. This strategy could reduce unexecuted order
volume by aggressive reduction of unexecuted order volume through reorders. What is more,
55
it is able to deal with the unexecution risk together with the timing risk. If some unexecuted
orders remain, they will be executed using market orders at maturity in order to achieve the
target. In general, market impact of market order is larger than that of limit order. The
proposing model is aimed at reducing execution cost through replacing limit orders not to
have unexecuted order volume at maturity. Trying to execute orders earlier could reduce
unexecuted order volume. This will be resulted in reducing timing risk because of the limited
effect of unpredictable price fluctuation.
Unexecuted order volume is defined as target execution volume,y, minus cumulative
executed order volume until the period,∑k
t wt . In the first period, unexecuted or remaining
order volume is equal to the target. On the other hand, when all limit orders and a market
order is posted, unexecuted order volume should be zero. This constraint shows that all target
is executed after using market order at maturity.
y(i )k = y(i )
k−1 − w(i )k (56)
y(i )0 = y (57)
y(i )N+1 = 0 (58)
Figure 9 illustrates overall image of this model.
56
Figure 9: Image of the Execution Strategy
The target order volume is defined asy and this amount is depicted as white circle. In the
first period, limit order volumez1 which is illustrated as blue bar will be posted. However,
a part of the posted order volume, which is illustrated as red allow, cannot be executed.
Executed order volume is illustrated asw1. The unexecuted order volumez1 − w1 is pooled
for a while. By adding this unexecuted order volume with the order volume which is not
placed in the first period, remaining order volumey1 can be defined.
In the second period, order volume will be decided by comparingx2 andy1. In the figure
9, y1 is assumed to be still large andx2 is chosen. The blue part of the circle fromy1 will be
placed asz2. As in the first period, red arrow part is not executed and this amount is pooled.
In the N-th period,yN−1 is assumed to be small enough and this is placed aszN. A part
of this amount cannot be executed which is shown asyN. At maturity (N+1-th period), all
57
unexecuted or remaining order volumeyN is placed as market order.
In the optimization model, the upper bounds of order volume in each periodxk will be
decided. If remaining unexecuted order volume before k-th period,yk−1, is smaller thanxk,
yk−1 will be placed. Otherwise, the unexecuted order volume is assumed to be large enough
andxk will be placed.
3.7.5 Execution Probability
Next, we model execution percentage and its probability (execution probability) for each
period. Basically, the probability of execution will decrease as the order volume increases.
However, posting limit orders could also induce opposite traders to post market orders. There-
fore, we assume that the execution probability does not depend on the order volume.
For probability distribution of execution percentageτ̃k, two distributions are defined. One
is the binomial distribution whose execution percentage takes only zero or one (no partial ex-
ecution). The other distribution is the truncated normal distribution. In this case, percentage
could take number between zero and one (partial execution). The mean execution probability
for these distributions are aligned. Examples of them are shown in Figure 10.
58
Binomial Ditribution
Fraction
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
05
1015
Truncated Normal Distribution
FractionD
ensi
ty
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Figure 10: Probability Distributions of Execution Volume
In the previous literatures of limit order strategy, the binomial distribution is only allowed.
However, if the order volume is large, this assumption is too strict. Large order volume can
be partially executed. Therefore, this research mainly discusses truncated normal distribution
case.
3.7.6 Price Dynamics and Market Impact Risk
The exogenous and fundamental process of stock price is assumed to follow the Brownian
motion. Usually, the stock price is defined as the Geometric Brownian motion if time span
is daily or monthly. However, the Geometric Brownian motion and the Arithmetic Brownian
motion are assumed to be almost the same in the short-term period such as intraday. Due to
its simplicity, we use the Arithmetic Brownian motion. Other effect which is endogenous is
59
price impact by limit and market order.
The price impact of limit order to executed order volumewk is modelled asLMI · wk.
LMI is the parameter for market impact of limit order. Unpredictable and exogenous price
change by other market participants is assumed to follow the normal distribution with mean
0 andσ volatility, (N(0, σ2)). Stock price at time k,Pk, is written as sum of the stock price
at k-1,Pk−1, the price impact by executed order volume, and the unpredictable price change.
In the first period, stock price is given byP0 which is not path dependent. This is shown in
the Formula (59).
Pk = Pk−1 + Pk−1 · (LMI · wk + σ · ξk) (k = 1, · · · , N) (59)
The unexecuted orders are executed using market orders at maturity, N. The price for
market order,PN+1, will be decided through market impact parameter of market order,MMI ,
and written in the Formula (60).
PN+1 = PN + PN · (MMI · wN+1 + σ · ξN+1) (60)
wN+1 is the amount of posted and executed market order at maturity. This volume is equal
to the amount that could not be executed through limit orders before the maturityy(i )N . Price
impact function for market order is also defined as linear relationship. Linear market impact
for market order is the same for previous literatures (Berstimas-Lo [1998], Almgren-Chriss
[2000]). This assumption is based on a strict assumption which the same amount of limit
order exists in all price levels of limit order book. Because of its simplicity, this research also
uses this concept.
60
3.7.7 Constraints of Execution Cost
The execution cost is defined as difference between the actual purchased price and the initial
price. The cumulative execution cost at k-th period is described in the Formula (61), which
is explained by the amount of executed orders,wk, multiplied by price difference between
the price after the price impact occurred,Pk and initial priceP0. The total execution cost is
defined as sum of the cumulative limit order cost and market order cost at maturity minus the
execution cost if all orders are assumed to be executed at the initial price.
C0 = 0 (61)
C(i )k = w(i )
k (P(i )k − P0) +C(i )
k−1 (k = 1, · · · , N + 1) (62)
3.7.8 Constraints of Target Executed Volume
These constraints show that sum of executed order volume through limit orders∑N
k=1 w(i )k and
market order volumew(i )N+1 achieves target volume y.
N+1∑k=1
w(i )k = y (63)
3.7.9 Constraints ofxt
In this model, there are no constraints for optimized posted order volumext . In case of
market order strategy, this constraint is included. This is because sum of posted order volume
is exactly equal to the target order volume. On the other side, limit order strategy which
allows to use reorder does not have this constraint. This is because there is unpredictability
in order execution. Due to this uncertainty, total posted order volume tends to be larger than
61
the target order volume. For example, when the expected execution probability is low, sum
of optimized posted order volume∑
xt will be larger. Otherwise, sum of the executed order
volume is far smaller than target volumey. By posting large order volume initially and post
unexecuted amount again in the later periods, the optimal strategy might try to use less market
order. Therefore, sum ofxt tends to be larger than total execution targety and constraints for
xt should not be included in the model.
62
4 Parameter Estimation
In this section, market impact parameters of limit order and market order (LMI andMMI ),
volatility of price change (σ), and execution probability (τk) will be estimated.
4.1 Literature Review
In this section, some previous literatures discussing parameter estimation are introduced. In
terms of market impact, one literature is done by Cont et al. [2013]. On the other hand,
as we have surveyed so far, there are three literatures which discuss estimation of execution
probability. Those literatures are Omura et al. [2000], Lo et al. [2002], and Cont-Kukanov
[2017].
4.1.1 Cont et al. [2013] (Estimation of Market Impact)
According to Cont et al. [2013], the Order Flow Imbalance (OFI) and price change are
linearly correlated. The OFI measures relative size of order inflow through comparing buy
and sell side of the limit order book. According to this literature, OFI in the period k is
defined as follows.
OFIk = Lbk(LimitBuyOrder) −Cbk(CancelBuyOrder) (64)
− Msk(MarketSellOrder) − Lsk(LimitSellOrder) (65)
+ Csk(CancelSellOrder) + Mbk(MarketBuyOrder) (66)
There are six elements in the OFI. The first three elements, limit buy (Lb), cancel buy
63
(Cb), and market sell orders (Ms) are buy side activities. Since cancel buy and market sell
orders consume liquidity of the buy side of the limit order book, these two activities have
downside pressure to the price. In other words, these two orders are negative sign for the
OFI. On the other hand, the last three terms, limit sell (Ls), cancel sell (Cs), and market buy
orders (Mb), are elements which occur in sell side of the limit order book. By comparing
buy side and sell side of the book, order flow imbalance is defined. In the later sections, this
assumption will be evaluated through tick by tick data analysis.
Using this definition, Cont et al. [2013] make a regression model to explain price change.
∆Pk = β ·OFIk + ϵ k (67)
Price difference∆Pk is defined as difference of mid-price between k-1 and k periods. Mid-
price is an average of the best bid and the best ask price. Mid-price can be computed by
Pk =Paskk+Pbid
k
2 . Paskk is the best ask price andPbid
k is the best bid price. If time interval is
15 minutes,∆Pk measures difference of mid-prices at the beginning of the period and at the
end of the period. This difference is called simple price difference which can be defined as
∆Pk = Pk − Pk−1.
In this research, the trade imbalance (T I) which is focusing on market order activities is
also defined. TheT I can be written as follows.
T Ik = Mbk − Msk (68)
ThisT I is also positively and linearly correlated with the price change. In this case, limit
and cancel order activities are included in the error term of regression model. Although this
notation captures transaction precisely, this literature confirms that relationship between price
64
change and theT I is weaker than that of theOFI. This is because the number of limit and
cancel order activities are much more than market orders. Since theT I model is noisier than
OFI model, this literature concludes to useOFI as an element to estimate price change.
For estimation, this literature uses trades and quotes data of fifty stocks from S&P 500. It
shows thatβ is positive and itsR2 is 0.691. The regression model is statistically significant.
That means, if limit buy, cancel sell, or market buy order inflow, price will change positively.
On the other hand, if cancel buy, market sell, or limit sell order come, these could cause
downside price change.
In this literature, market impact is thought to be linear to price change. This means that
depth of the limit order book is the same size for all price levels. In other words, limit order
book is flat shaped.
In the later sections, this regression model will be examined and modified so that param-
eter in the optimization model can be estimated.
4.1.2 Omura et al. [2000] (Estimation of Execution probability)
In the research done by Omura et al. [2000], execution probability of limit orders is estimated
using limit order book data of the Tokyo Stock Exchange in December, 1998. The assets of
Tokyo Stock Exchange are divided into five groups by monthly trading volume. Among
assets in one group, top ten companies are selected as representatives of the group. In this
literature, order flow is constructed in order to estimate execution probability.
The best bid price at period t is defined asBidt . Limit buy order volume at period t to
price b is defined asBLmtbt . BidVbt stands for depth at the best bid at period t of price b.
65
BLmtbt , is defined as follows.
i f Bidt = Bidt−1 : BLmtbt = BidVbt − BidVb
t−1 (69)
i f Bidt < Bidt−1 : BLmtbt = 0 (70)
i f Bidt > Bidt−1 : BLmtbt = BidVbt (71)
If the best price is not changed during the period (Bidt = Bidt−1), difference of the best
bid depth between t-1 and t (BLmtbt − BLmtbt−1) is thought to be total order flow of limit buy
order. Although some limit orders are cancelled or some others are consumed by market sell
orders, simple difference of the best bid depth is assumed to be order flow in buy side. For
example, if new limit order volume is large,BidVbt is much greater thanBidVb
t−1. Therefore,
BLmtbt is positive. On the other hand, if new limit order volume is small limit order volume
which is consumed through cancel and market orders are larger,BidVbt−1 is larger thanBidVb
t
andBLmtbt is negative. This means that order flow is negative and outflow exceeds inflow.
In this research, the dataset includes only information of the best price at the moment.
They cannot track order inflow if the best price is changed positively. Therefore, incoming
limit orders in caseBidt < Bidt−1 is set to be zero.
On the other hand, if the best bid price is increased (Bidt > Bidt−1), order inflow is
assumed to be equal to new depth in the new price level. This case means new order is
posted above the best price and the best price is renewed. Due to the same reason with
Bidt < Bidt−1 case, depth information of the new price level at the t-1 period is not available
and is not considered in the model. Therefore, inflow to the previous best bid depth at t-1
66
period is ignored and only depth at the new best price is considered. Using these definitions,
execution probability is defined as follows.
BBookt ≤T∑
s=t+1
BMkts (72)
BBookt = BBookt−1 + BLmtt (73)
BBook means remaining limit buy orders in the limit order book at period t.BMktt
means inflowing market order volume at period t. If the incoming market order volume
from the next period is larger than remaining order volume, the limit order is assumed to be
executed. However, in this model, all limit orders posted at period t is assumed not to be
cancelled until maturity.
4.1.3 Lo et al. [2002] (Estimation of Execution Probability)
The next literature by Lo et al. [2002] discusses execution probability from price dynamics
side. In this literature, limit buy order is executed if and only if the best price becomes equal
to or lower than the initial best price. LetBidmin denote the minimum best bid price during a
period. Execution probability is defined as follows.
i f Bidmin ≤ Bidt : Probability = 1.0 (74)
i f Bidt < Bidmin : Probability = 0.0 (75)
More precisely, a buy limit order which is posted to the priceBidt0 at time t0 will be
executed in the interval [t0, t0 + t] if and only if the minimum of the best bid price during
67
the interval t,Bidmin, is less than or equal toBidt0. If the Bidmin is higher thanBidt0,
the best bid price increases during the interval, all posted limit orders are not able to be
executed, and execution probability is assumed to be zero. According to this literature, if
price dynamics is defined as the Geometric Brownian motion, probability of the limit order
to be filled is computed as in the following equations. These equations describe probability
of the minimum of the best bid priceBidmin lower than or equal to the initial best priceBidt
if the initial best priceBid(t0) is equal toBid0.
dP(t) = αP(t)dt + σP(t)dW(t)
Prob{Pmin ≤ Pt |P(t0) = P0} = 1− Φ(log(P0/Pt ) + µt
σ√
t)
+ (Pt
P0)
2µσ2Φ(
log(Pt/P0) + µt
σ√
t)
µ = α − 12σ2
This definition is able to interpret the case when the best bid price is completely consumed
by incoming market order and the best price changes. However, it cannot compute the case
when trader cancels and replaces the limit order due to holding enough execution probability.
4.1.4 Cont-Kukanov [2017] (Estimation of Execution Probability)
Finally, Cont-Kukanov [2017] illustrate another model to estimate execution probability. In
this literature, filled order volume is defined as in the following equations.
min(max(ξ −Q, 0), L) = max(ξ −Q, 0) −max(ξ −Q − L, 0) (76)
68
Figure 11: Filled Amount in Cont-Kukanov [2017] Model
(Source: Cont-Kukanov [2017])
L is limit order volume posted by the trader andQ is the order queue which is posted
by other traders before the submission ofL. During the period, cancel orders and market
orders are posted which are defined asξ = Cancel(C) + Market(D). If the incomingξ is
larger than order queue (Q), the excess amount is compared with limit orderL. If the excess
amount is higher thanL, all L is executed whereasL is partially filled if the excess amount is
smaller thanL. On the other hand, if the incomingξ is smaller than order queueQ, Q is not
executed completely and limit orderL is not executed at all. Although this model considers
all three order types which are limit, cancel, and market orders, cancel orders are assumed
to be targeted only to the order queue and not to the new limit orders. Since time interval is
assumed to be one minute in this literature, this assumption can be held. However, if the time
interval is widened such as 15 minutes, cancel order is also assumed to be targeted to the new
limit orders. And also, price dynamics is not included and the best price is assumed to be the
69
same during the period. These constraints are relatively strong and should be modified for
this research.
70
4.2 Objectives, Research Methodology, and Research Framework
In this section, objectives, research methodology, and research framework of parameter es-
timation are illustrated. First, dataset to be used is introduced. Then, model for estimating
market impact parameters will be described. Finally, estimation model for execution proba-
bility is introduced.
4.2.1 Dataset
In this analysis, tick by tick data of Tokyo Stock Exchange in 2012 from January 4th to June
29th is used to estimate parameters. Among assets in the Tokyo Stock Exchange, Nikkei 225
(Nikkei Stock Average) stocks are extracted. According to Nikkei [2017], the Nikkei 225 is
one of the most famous stock average consisted of 225 assets listed in the 1st section of the
Tokyo Stock Exchange.
In the dataset, price and depth up to eighth level for both bid and ask side are available.
This depth data is updated every inflow of limit, cancel, or market orders. We do not use all
assets in the Nikkei 225 lists for simplicity. Among listed stocks, following criteria are used
to decide target companies to be investigated.
• Listed in 2012 continuously
• Quote is one yen and is not changed in 2012
• Stock price is below 3000 yen
These criteria are decided in order to eliminate effects of IPO, price range, and other activ-
ities. Among the Nikkei 225 stocks, 196 stocks have these characteristics. These 196 stocks
71
are grouped into four groups, 49 stocks in each group, ordered by turnovers. For simplicity,
four chosen assets are labeled as A to D and these assets are summarized in Table 5. The
four assets are Bank of Mitsubishi Tokyo UFJ, Ltd. (Code: 8306), Nippon Yusen Kabushiki
Kaisha (Code: 9101), Mitsui Chemicals, Inc. (Code: 4183), and MinebeaMitsumi Inc. (Code
6479). These four stocks are lined in liquidity order. The most liquid asset is stock A (Bank
of Mitsubishi Tokyo UFJ) and the least liquid asset is stock D (MinebeaMitsumi). LO stands
for limit order and MO stands for market order. Daily change, value, volume, count, and
all values are average using six months data. The table illustrates average number of price
changes in a day, average daily limit order volume, average daily market order volume, aver-
age daily limit order value, average daily market order value, average number of daily limit
orders, average number of daily market orders, volatility of price change measured by using
log difference, and mean of closing price. Value is calculated through multiplying volume
and asset price.
The time series data is divided into 15 minutes intervals from 9:00 to 11:30 and 12:30
to 15:00. In order to eliminate effect of Itayose, sessions of very first in the morning and in
the late afternoon are omitted. The Itayose is a system at the Tokyo Stock Exchange used
to determine opening price. First, all posted orders are pooled in the limit order book. After
that, market order is prioritized and executed. Next, limit buy orders posted to higher price
and limit sell orders posted to lower price are matched. Therefore, 9:15 to 11:30 and 12:30
to 14:45 data when the orders are continuously matched is used.
72
Table 5: Four Assets and Their Codes
A B C D
Asset Code 8306 9101 4183 6479
Company Name Mitsubishi UFJ Nippon Yusen Mitsui Chemicals MinebeaMitsumi
Daily Price Change 303 201 172 275
(times/day)
Daily LO Volume 87,458,638 33,182,176 19,790,975 6,861,603
(volume/day)
Daily MO Volume 47,250,194 15,415,843 8,146,820 2,767,355
(volume/day)
Daily LO Value 33,370,954,368 7,382,986,667 4,642,739,008 2,385,448,347
(yen/day)
Daily MO Value 18,120,300,718 3,441,354,213 1,931,790,779 967,024,868
(yen/day)
Daily LO Count 11,923 4,074 3,498 2,503
(times/day)
Daily MO Count 4,619 1,227 1,023 665
(times/day)
Mean Closing Price 378.08 219.67 234.85 348.79
(LO: Limit Order/MO: Market Order)
73
From next section, four companies are written as stock A to stock D.
4.2.2 Estimation of Market Impact
For market impact estimation, some models are compared with the previous literature model
by Cont et al. [2013].
Model 0 is using previous literature framework. This model explains price difference
using the order flow imbalance (OFI). In the OFI, limit, cancel, and market orders are equally
considered. Price change is defined as absolute difference. The model 0 uses assumption
that all six elements of the OFI are equally influential to price change. Through comparing
with other models, validity of this assumption will be discussed. Definition of the OFI and
regression model is described below.
OFIk = Lbk −Cbk − Msk − Lsk +Csk + Mbk (77)
∆Pk = α + β ·OFIk + ϵ k (78)
In the model 1, the OFI elements are decomposed. Limit buy, cancel buy, market sell,
limit sell, cancel sell, and market buy orders are used independently in regression model. In
this model, price change is defined as log difference. And also, limit, cancel, and market
orders are measured as order value. By including price information and measuring log price
difference or percentage price difference, four stocks are able to be compared without any
transformation. By discussing result of this model 1, assumption of signs in the OFI elements
and assumption that all six elements are equally influential will be verified. Definition of the
74
model 1 is described as follows.
∆Pt = logPt − logPt−1 (orPt − Pt−1
Pt−1) (79)
= α + β1Lbt + β2Cbt + β3Mst + β4Lst + β5Cst + β6Mbt (80)
Although the model 1 is able to consider OFI elements separately, some of them are highly
correlated. And also, since the optimization model uses limit orders and market orders, cancel
order could be eliminated from regression. Therefore, after discussing collinearity among six
elements of OFI, this model 2 will be introduced. The model 2 is defined as follows.
∆Pt = α + β1Lb + β2Ms + β3Ls + β4Mb (81)
75
4.2.3 Estimation of Price Volatility
Price volatility will be estimated using three definitions. The first definition uses absolute
difference which is also used in the previous literature. The second definition uses percent-
age difference. The last one is log difference. Log difference and percentage difference are
assumed to be almost similar.
• ∆P1t = Pt − Pt−1
• ∆P2t =
Pt−Pt−1Pt−1
• ∆P3t = log Pt
Pt−1= logPt − logPt−1
If the first absolute difference is used, this cannot be comparable since price ranges are
different among four stocks. For example, standard deviation could be large if the price range
is large. This does not reflect degree of liquidity. In general, volatility of price is assumed
to be small in case of high liquid asset and large in case of low liquid asset. Therefore, by
considering percentage difference or log difference, four assets are able to be compared from
perspective of liquidity.
76
4.2.4 Estimation of Execution Probability
By referring to the models of execution probability in the previous literatures, new estimation
model is defined. In this research, time interval is set to be 15 minutes which is relatively
long compared with the high frequency trading. Therefore, the posted limit order during the
interval could be cancelled and replaced. Main difference from previous literature is that
cancel order targets both new posted limit orders and already existing depth.
LOR(Limit Order Ratio) =Lbt
Lbt + Vt−1
1. Pt−1 ≤ Pmin : ExeProbt = min(max(Mst − (Vt−1 −Cbt · (1− LOR))
Lbt −Cbt · LOR, 0), 1)
2. Pmin < Pt−1 : ExeProbt = 1
Lb is limit buy order volume to the best price and Cb is cancel buy order volume to the
best price. V means the best bid depth. The LOR stands for the limit order ratio which
measures relative volume of incoming new limit order compared with the best bid depth at
the beginning of the period.
Through considering new posted limit order volume and cancel order which targets the
new limit order,Lbt − Cbt · LORcalculates new limit order volume which is not cancelled
during the interval. This volume will be compared with executed order volume.Mst− (Vt−1−
Cbt ·(1−LOR)) describes incoming market order volume minus depth which is not cancelled.
In this model, target of cancel orders is both new limit orders and best bid depth already
exists. Table 6 shows relative volume of order flow elements compared with the depth at the
beginning of the period in case of stock A (8306). Limit buy, cancel buy, and market sell
77
order volumes are compared with the best bid depth (BD). Limit sell, cancel sell, and market
sell order volumes, on the other hand, are compared with the best ask depth (AD).
Table 6: Relative Volume of OFI Elements to Depth (Stock A, 8306)
Lb_BD Cb_BD Ms_BD Ls_AD Cs_AD Ms_AD
Min. 0.044 0.020 0.012 0.060 0.023 0.010
1st Qu. 1.075 0.594 0.374 1.160 0.578 0.366
Median 2.117 1.206 1.118 2.154 1.133 1.151
Mean 3.355 1.949 1.719 3.897 1.983 1.940
3rd Qu. 3.849 2.215 2.002 3.856 2.051 2.171
Max. 97.047 86.456 48.766 1124.579 310.579 361.702
Since the depth fluctuates continuously, distribution of relative volumes of OFI elements
have long tail in the large side. Although median and minimum is close, maximum is very
far from median. In other words, in some periods, depth is very small and inflowing order
volume is so large that relative volume of inflow is hundreds and thousands more than depth.
In general, by comparing market order volume and depth, these two volumes are almost the
same. If all depth volumes are assumed not to be cancelled during the interval and the best
price is assumed not to be changed, any new posted orders are not executed. Therefore, depth
cannot be ignored in order to consider execution probability. In this research, relative volume
of cancel order targeting new coming limit orders is assumed to be the same with relative
volume of new limit orders compared with depth size. By using limit order ratio (LOR),
which explains relative volume of limit buy orders compared with depth size, cancel order
78
volume is divided into two.
On the other hand, when the best bid price during the interval goes below the initial
best bid price, the best bid depth is completely consumed through cancel and market orders.
Therefore, all orders which are posted to the previous best price are assumed to be executed
completely in this case. Therefore, execution probability becomes one ifPmin goes below the
best price in the beginning (Pt−1).
For all four stocks, median of the relative ratio of OFI elements are summarized in Figure
12.
!
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Figure 12: Relative Percentage of OFI Elements to Depth (All Stocks)
For all six elements, depth is almost the same or slightly smaller than cancel or market
order volumes. Limit order volume is around twice larger than depth size. Although stock A
to C are almost the same results, stock D is relatively higher than others. This might be due
79
to less incentive to make depth. If relative percentage of OFI elements is high, this is because
of large coming order volumes or small depth.
Therefore, depth cannot be ignored in order to consider cancel orders. In conclusion,
assumption that cancel order targets both previously pooled depth and new limit orders is
reasonable if the time interval is 15 minutes.
80
4.3 Findings, Discussion, and Conclusions
In this section, parameter estimation models are implemented and their findings, discussion,
and conclusions are discussed. First, elements of the Order Flow Imbalance (OFI) are sum-
marized. Then, volatility of price change,σ, will be estimated. After that, regression model
to estimate price impact will be calculated under some model settings. Finally, execution
probability is estimated.
4.3.1 Sum of Order Size
First, sum of order size is examined. This measures how many orders there are coming in 15
minutes time interval. This is measured as volume. In case of stock A (8306, Bank of Tokyo
Mitsubishi UFJ), sum of order size distribution is summarized in Table 7.
81
Table 7: Distribution of Sum of Order Size (Stock A, Interval=15min)
Sum Lb Cb Ms Ls Cs Mb
Min. 85,000 32,500 31,300 127,300 31,200 14,800
1st Qu. 1,175,000 654,200 428,000 1,356,000 640,500 400,700
Median 2,082,000 1,172,000 1,002,000 2,210,000 1,135,000 1,074,000
Mean 2,402,000 1,368,000 1,258,000 2,456,000 1,327,000 1,367,000
3rd Qu. 3,220,000 1,854,000 1,737,000 3,261,000 1,797,000 1,901,000
Max. 14,760,000 7,035,000 8,124,000 9,972,000 7,336,000 9,478,000
sd 1,656,976 962,479 1,080,414 1,508,115 927,550 1,267,133
skewness 1.592 1.283 1.685 1.127 1.363 1.893
kurtosis 7.981 5.502 7.375 4.783 6.188 8.410
Box plot of this distribution is drawn in Figure 13.
82
Figure 13: Boxplot of Sum of Order Size (Stock A, Interval=15min)
According to the Table 7 and the Figure 13, limit order size is almost twice larger than
cancel and market orders. On the other hand, market orders are the least size. However,
difference between market and cancel orders are small and these two order types are almost
the same size. By looking at the figure, skewness, and kurtosis, distribution is fat tail in the
right side. That means, only small number of periods have extremely large order inflow. In
the majority of periods, inflowing order size is stable.
Correlation matrix of these variables is calculated in Table 8.
83
Table 8: Correlation Matrix of Sum of Order Size (Stock A, Interval=15min)
Lb Cb Ms Ls Cs Mb
Lb 1.000 0.759 0.504 0.462 0.600 0.751
Cb 0.759 1.000 0.541 0.549 0.471 0.434
Ms 0.504 0.541 1.000 0.729 0.422 0.489
Ls 0.462 0.549 0.729 1.000 0.722 0.544
Cs 0.600 0.471 0.422 0.722 1.000 0.557
Mb 0.751 0.434 0.489 0.544 0.557 1.000
Limit and cancel orders at the same side of the limit order book have high correlations.
For example, limit buy and cancel buy orders are highly correlated. This might be because
many investors cancel posted limit order and replace them with the same size to new price
levels in order to have appropriate execution probability. And also, market and limit orders of
the same direction such as limit buy and market buy have high correlations. This is assumed
to be occurred because investors give up using limit orders and use market orders to obtain
certainty of execution.
By summarizing all four assets, medians of sum of order size are summarized in Figure
14.
84
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+,-./0.12342.+564
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Figure 14: Medians of Sum of Order Size (All Stocks, Interval=15min)
For other four stocks, distributions among three order types are almost the same. The
largest is limit order, the second largest is cancel order, and the least is market order. However,
difference between cancel and market orders is not small in case of stock B, C, and D. In
general, limit orders are twice larger than cancel and market order volume. This shows that
size of order inflow and order outflow is almost balanced and market is stable in the long run.
By focusing on differences among four stocks, order size of stock A is far larger than other
three assets.
4.3.2 Average Order Size
Next, average order size is calculated. In 15 minutes interval, all coming orders are checked
and average size of those orders is calculated. Distribution of stock A is summarized in Table
85
9.
Table 9: Distribution of Average Order Size (Stock A, Interval=15min)
Avg Lb Cb Ms Ls Cs Mb
Min. 571 224 256 708 277 127
1st Qu. 2,800 2,028 1,566 2,998 2,075 1,461
Median 3,936 3,352 2,714 4,109 3,414 2,709
Mean 4,324 4,096 3,547 4,642 4,236 3,772
3rd Qu. 5,342 5,293 4,511 5,708 5,417 4,616
Max. 26,830 35,850 27,400 38,220 30,130 44,650
sd 2,253 3,065 3,087 2,600 3,190 3,852
skewness 2.110 2.547 2.738 3.148 2.274 3.491
kurtosis 14.150 16.884 14.624 28.584 12.271 23.522
Box plot of this distribution is illustrated in Figure 15.
86
Figure 15: Distribution of Average Order Size (Stock A, Interval=15min)
As in the previous discussion of sum of order size, limit orders are larger than cancel
and market orders. However, difference is smaller compared with the last analysis. In case
of sum of order size, medians of limit orders are almost twice larger than other two order
types. However, average order size of limit order is not twice larger than others. By looking
at maximum or larger values, six order types are almost the same. Therefore, size of single
order is avoided to be extremely large. Because of market impact, large order might be
avoided. Right skewness is the same with previous result.
Correlation matrix of these variables is summarized in the Table 10.
87
Table 10: Correlation of Average Order Size (Stock A, Interval=15min)
Lb Cb Ms Ls Cs Mb
Lb 1.000 0.762 0.261 0.162 0.218 0.247
Cb 0.762 1.000 0.245 0.163 0.187 0.198
Ms 0.261 0.245 1.000 0.228 0.243 0.358
Ls 0.162 0.163 0.228 1.000 0.660 0.419
Cs 0.218 0.187 0.243 0.660 1.000 0.227
Mb 0.247 0.198 0.358 0.419 0.227 1.000
Compared with sum of order size, average order sizes are not highly correlated except
limit and cancel orders of the same side such as limit buy and cancel buy, or limit sell and
cancel sell. This might be traders cancel previous limit order and replace it to the price where
execution is easier. Therefore, cancel and limit orders are connected and highly correlated
even in case of average size.
For other four assets, medians of average order size are summarized in Figure 16.
88
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Figure 16: Distribution of Average Order Size (All Stocks, Interval=15min)
Characteristics are almost the same. Limit order is the largest and other two order types
are smaller. However, difference among four assets becomes smaller. Therefore, average
order size does not explain difference of sum of order size completely. By considering both
order size and number of orders, difference of sum of order size can be explained. Number
of orders will be discussed in the later section.
4.3.3 Standard Deviation of Order Size
Next, standard deviation of order size is compared. All orders which inflow during inter-
vals of 15 minutes are checked and their standard deviations are compared. Distribution of
standard deviation in case of stock A is summarized in Table 11 and Figure 17.
89
Table 11: Distribution of Standard Deviation of Order Size (Stock A, Interval=15min)
Sd Size Lb Cb Ms Ls Cs Mb
Min. 14,750 4,406 990 17,580 96 1,061
1st Qu. 306,200 144,800 90,410 342,900 143,500 87,610
Median 480,400 275,500 204,800 513,200 279,500 213,100
Mean 541,400 338,900 291,200 576,200 339,500 318,600
3rd Qu. 709,000 461,900 396,000 735,500 469,400 433,000
Max. 2,819,000 2,063,000 2,282,000 4,472,000 1,987,000 3,763,000
sd 329,557 264,120 288,832 347,120 267,123 344,591
skewness 1.400 1.430 2.169 2.272 1.592 2.818
kurtosis 6.699 5.831 9.591 18.449 6.912 17.523
Box plot of this distribution is drawn in Figure 17.
90
Figure 17: Distribution of Standard Deviation of Order Size (Stovk A, Interval=15min)
According to this distribution, limit orders are highly deviated compared with other order
types. Limit orders are around twice larger standard deviation than cancel or market orders.
As in the previous cases, distribution is right skewed.
In case of other stocks, median of standard deviation is summarized in Figure 18.
91
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Figure 18: Distribution of Standard Deviation of Order Size (All Stocks, Interval=15min)
In all stocks, limit orders have the largest medians of standard deviations, cancel orders
are the second largest, and market orders have the least standard deviations.
4.3.4 Order Value
In this part, order value is compared. Order value is defined using price and executed order
size. In case of Tokyo Stock Exchange market, this value is measured as yen.
OrderValue= BestPrice·OrderSize (82)
According to this definition, distribution of order value for stock A, 8306, is summarized
in Table 12. Since order value is very large, numbers are standardized into per one million
yen.
92
Table 12: Distribution of Order Value (Stock A, Interval=15min)
Volume Lb Cb Ms Ls Cs Mb
Min. 28.390 10.860 10.420 43.030 10.360 5.002
1st Qu. 439.600 247.600 155.400 498.100 239.400 151.100
Median 785.500 448.600 383.700 831.200 429.900 407.000
Mean 914.700 521.500 482.300 939.300 507.600 524.400
3rd Qu. 1,240.000 714.300 667.600 1,243.000 681.300 726.500
Max. 5,940.000 2,845.000 3,352.000 4,053.000 2,853.000 3,834.000
sd 639.435 375.622 426.391 598.001 365.231 493.652
skewness 1.480 1.347 1.782 1.189 1.424 1.893
kurtosis 6.945 5.771 7.940 4.926 6.389 8.310
Box plot of this distribution is Figure 19.
93
Figure 19: Distribution of Order Value (Stock A, Interval=15min)
Limit orders are relatively high sum of order values. Other two order types are relatively
lower order values. Market orders are relatively higher standard deviations than cancel orders.
Correlation Matrix of order value is calculated in Table 13.
94
Table 13: Correlation Matrix of Order Value (Stock A, Interval=15min)
Lb Cb Ms Ls Cs Mb
Lb 1.000 0.766 0.529 0.504 0.624 0.765
Cb 0.766 1.000 0.570 0.585 0.499 0.462
Ms 0.529 0.570 1.000 0.745 0.454 0.510
Ls 0.504 0.585 0.745 1.000 0.742 0.562
Cs 0.624 0.499 0.454 0.742 1.000 0.576
Mb 0.765 0.462 0.510 0.562 0.576 1.000
The result is almost the same with sum of order size. Limit and cancel orders are relatively
high correlation. And also, limit and market orders of the same side of the limit order book
are highly correlated.
For other stocks, median of order value is summarized in Figure 20.
95
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Figure 20: Distribution of Order Value (All Stocks, Interval=15min)
This result is almost the same with sum order size. Limit orders are the largest and market
orders are the least. By focusing on difference among four stocks, stock A is far larger than
other three assets.
4.3.5 The Number of Orders
Finally, the number of orders is compared. Here, size or value of each single order is ignored.
Distribution of stock A is summarized in Table 14.
96
Table 14: Distribution of the Number of Orders (Stock A, Interval=15min)
Number Lb Cb Ms Ls Cs Mb
Min. 50 29 23 61 17 15
1st Qu. 213 113 82 224 113 81
Median 298 160 112 304 155 115
Mean 327 178 129 336 176 127
3rd Qu. 411 222 157 409 214 160
Max. 1,091 789 562 8,767 8,766 666
sd 151 90 71 236 205 65
skewness 1.122 1.438 1.820 21.430 34.370 1.539
kurtosis 4.913 6.714 7.915 755.641 1437.711 7.800
Box plot of this distribution is illustrated in Figure 21.
97
Figure 21: Distribution of the Number of Orders (Stock A, Interval=15min)
Although a few outliers exist, distribution is almost the same with others. Limit orders
have the largest number of orders and market orders have the least number of orders. These
outliers in the number of limit sell and cancel sell orders occurs at the same date. On June
7th, around 11:26 to 11:29, many orders which is 200 volumes of limit sell and cancel sell
are posted continuously. Although this trade exists in the dataset, it is assumed to be irregular
pattern. For example, this is assumed to be caused by algorithmic error. The next Table 15
excludes these outliers.
98
Table 15: Distribution of Number of Orders without Outliers (Stock A, Interval=15min)
Number Lb Cb Ms Ls Cs Mb
Min. 50 29 23 61 17 15
1st Qu. 213 113 82 224 113 81
Median 298 160 112 304 155 115
Mean 327 178 129 332 172 127
3rd Qu. 411 222 157 409 214 160
Max. 1,091 789 562 1,271 1,114 666
sd 151 90 71 151 88 65
skewness 1.124 1.440 1.819 1.257 1.764 1.538
kurtosis 4.916 6.728 7.911 5.580 11.233 7.799
Box plot will be modified without outliers. It is described in Figure 22.
99
Figure 22: Distribution of Order Volume without Outliers (Stock A, Interval=15min)
By eliminating outliers, distributions are cleared. Correlation Matrix of these variables
before eliminating outliers is calculated in Table 16.
100
Table 16: Correlation Matrix of Number of Orders(Stock A, Interval=15min)
Lb Cb Ms Ls Cs Mb
Lb 1.000 0.639 0.431 0.329 0.246 0.630
Cb 0.639 1.000 0.421 0.407 0.271 0.422
Ms 0.431 0.421 1.000 0.368 0.153 0.513
Ls 0.329 0.407 0.368 1.000 0.866 0.273
Cs 0.246 0.271 0.153 0.866 1.000 0.170
Mb 0.630 0.422 0.513 0.273 0.170 1.000
Limit sell and cancel sell orders are especially low correlations. Although other four order
types are around 0.5 in correlation, these two orders are around 0.2. Correlation of limit
sell and cancel sell orders is 0.866 and this is unusually high. Due to these reasons, these
correlations are highly influenced by the outlier. Therefore, this correlation matrix should
be calculated again without outliers. After eliminating outliers, new correlation matrix is
illustrated in Table 17.
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Table 17: Correlation Matrix of Number of Orders without Outliers (Stock A, Inter-
val=15min)
Lb Cb Ms Ls Cs Mb
Lb 1.000 0.639 0.431 0.491 0.530 0.630
Cb 0.639 1.000 0.421 0.597 0.561 0.423
Ms 0.431 0.421 1.000 0.575 0.357 0.513
Ls 0.491 0.597 0.575 1.000 0.624 0.449
Cs 0.530 0.561 0.357 0.624 1.000 0.433
Mb 0.630 0.423 0.513 0.449 0.433 1.000
By eliminating outliers, correlation becomes similar to the findings of previous literature.
As in the previous settings, limit and cancel orders of the same side are highly correlated.
Market and limit orders of the same directions are also high correlations.
In case of other stocks, distribution for number of orders is summarized in Figure 23.
102
!
"!
#!!
#"!
$!!
$"!
%!!
%"!
&' (' )* &* (* )'
+,-./01'-2.34.526-2*
7 8 ( 9
Figure 23: Distribution of Order Volume (All Stocks, Interval=15min)
Distribution is almost the same with other cases.
103
4.3.6 Estimation of Price Volatility σ
In this section, volatility of price change,σ, will be estimated. Before doing estimation, plots
of mid-price for six months for four stocks are illustrated in Figure 24 to Figure 27. Hori-
zontal axis shows time which is measured as interval of 15 minutes. Vertical axis illustrates
price.
Figure 24: Price Dynamics (Stock A)
104
Figure 25: Price Dynamics (Stock B)
Figure 26: Price Dynamics (Stock C)
105
Figure 27: Price Dynamics (Stock D)
Based on definitions introduced in the previous section, standard deviation of the price
change is computed using all six months data. Three definitions are absolute difference,
percentage difference, and log difference. In order to calculate daily volatility, following
transformation has been done. There are 18 periods of 15 minutes in a day.
σ1day =
√(Number o f Periods/day) · σ2
15min (83)
Result is summarized in Table 18.
106
Table 18: Result of Price Volatilityσ Estimation
A B C D
P1t (Absolute Diff) 1.0619 0.8877 0.8604 1.4366
P2t (Percentage Diff) 0.0026 0.0041 0.0037 0.0041
P3t (Log Diff) 0.0026 0.0041 0.0037 0.0041
P2t (Daily, Percentage) 0.0112 0.0172 0.0156 0.0175
P3t (Daily, Log) 0.0112 0.0172 0.0156 0.0175
Mid-price Range 180.5-284.5 296.5-400 325.5-447.5 172.5-270.5
According to this result, stock D has the largest standard deviation. Although stock A is
the second largest inP1t , it is the least in other two definitions. This is due to the price range.
Since price range of stock A is higher than stock B and stock C, standard deviation of stock
A under definition of absolute difference becomes larger than others. However, this does not
imply difference of liquidity. Therefore, percentage and log differences are also compared.
When focusing on percentage and log differences, order of volatility among four assets are
almost the same with order of liquidity. More precisely, in case of stock A, liquidity has the
largest and volatility of price change is the least. On the contrary, stock D, which is the least
liquid asset, is the largest volatility. In case of one day, mid-price is changed around 1.5% on
average.
107
4.3.7 The Order Flow Imbalance and Price Change
Before estimating Market Impact, relationship between Order Flow Imbalance (OFI) and
price change will be examined. According to the previous literature, OFI is defined as fol-
lows.
OFIk = Lbk −Cb
k − Msk − Ls
k +Csk + Mb
k (84)
Price change can be defined using absolute difference, percentage difference, and log
difference. At first, absolute price difference is used to clarify the relationship between OFI
and price change. Relationship of stock A is described in Figure 28.
Figure 28: Absolute Price Change and OFI (Stock A, Absolute Price Difference)
Figure 28 shows linear relationship between OFI to x axis and Price Change to y axis. If
the OFI is negative, price will be changed negatively and decrease. Therefore, linear regres-
108
sion model in the next sections might be able to explain this relationship.
As alternative definitions, percentage and log price differences are also discussed. OFI
and percentage difference are depicted in Figure 29.
Figure 29: Percentage Price Change and OFI (Stock A, Percentage Price Difference)
As in the absolute difference setting, OFI and percentage price change are positively and
linearly correlated.
Finally, OFI and price change measured as log difference are summarized in Figure 30.
109
Figure 30: Log Price Change and OFI (Stock A, Log Price Difference)
Relationship is almost the same with percentage difference. In any cases, price difference
and OFI are characterized as linear and positive relationship.
Next, market order and other two order types are separately considered.T I stands for
trade imbalance which considers market orders andOFIWOTI stands for the Order Flow
Imbalance without Trade Imbalance which considers other two order types. Using following
definitions, relationships between price change and OFI elements are shown in Figure 31 and
Figure 32.
T Ik = Mbk − Ms
k (85)
OFIWOTIk = Lbk −Cb
k − Lsk +Cs
k (86)
110
Figure 31: Absolute Price Change and TI (Stock A)
111
Figure 32: Absolute Price Change and OFIWOTI (Stock A)
Figure 31 and Figure 32 show that both market order and other order elements have
positive and linear relationship with price change.
112
4.3.8 Estimation of Market Impact: Model 0
First, result of model 0 which uses framework from previous literature by Cont et al. [2013] is
discussed. In order to measure price change, absolute difference, percentage difference, and
log difference are used. The previous literature introduced before uses absolute price differ-
ence. For independent variable, order value is used. Definition of this model 0 is illustrated
below.
∆Pt = α + β ·OFIt (87)
First, result under absolute difference setting is summarized in Table 19. Since order
value is very large numbers, the value is standardized into per one million yen.
Table 19: Result of Model 0 Regression (per 1 million yen, Absolute Price Difference)
Stock Estimate Std. Error t value Pr(>|t|)
A (Intercept) -0.00382 0.00948 -0.403 0.687
β 0.00094 0.00001 94.863<2e-16
B (Intercept) 0.00847 0.01007 0.840 0.401
β 0.00354 0.00005 76.220<2e-16
C (Intercept) -0.01086 0.00954 -1.138 0.255
β 0.00628 0.00008 77.039<2e-16
D (Intercept) 0.02635 0.01766 1.492 0.136
β 0.02104 0.00032 66.685<2e-16
According to this result, OFI is able to explain absolute price change to some extent. The
113
coefficient of OFI is positive. This implies that if order flow is positive, price change is also
positive. On the other hand, t value and p value show that intercepts of all four assets are not
statistically significant. Hypothesis that intercepts are zero cannot be rejected. By comparing
among four assets, the coefficient is small in case of stock A, which is the most liquid asset.
If its liquidity is low such as stock D, market impact coefficient becomes larger.
Other results of this regression model such as f statistics are summarized in Table 20.
Table 20: Other Results of Model 0 (Absolute Price Difference)
A B C D
Residual Std. Error 0.4407 0.444 0.4471 0.8236
df 2168 1942 2194 2176
F-statistic 8999 5809 5935 4447
df 1, 2158 1, 1942 1, 2194 1, 2176
p-value <2.2e-16 <2.2e-16 <2.2e-16 <2.2e-16
R-squared 0.8066 0.7094 0.7301 0.6714
Adj. R-squared 0.8065 0.7493 0.7300 0.6713
AIC 2594.162 2363.985 2700.402 5339.772
BIC 2611.195 2380.702 2717.485 5336.83
The p-value of testing that each parameter is equal to zero shows that the hypothesis is
rejected and the regression is statistically significant. R-squared and Adjusted R-squared are
also relatively large.
If using percentage difference, estimated coefficients are calculated in Table 21.
114
Table 21: Result of Model 0 Regression (per 1 million yen, Percentage Price Difference)
Stock Estimate Std. Error t value Pr(>|t|)
A (Intercept) -0.000001 0.000026 -0.031 0.975
β 0.000002 0.000000 90.800<2e-16
B (Intercept) 0.000050 0.000048 1.034 0.301
β 0.000016 0.000000 72.500<2e-16
C (Intercept) -0.000039 0.000042 -0.911 0.362
β 0.000026 0.000000 72.579<2e-16
D (Intercept) 0.000090 0.000051 1.774 0.0763
β 0.000060 0.000001 66.550<2e-16
In case of log price difference, Table 22 shows estimated coefficients.
115
Table 22: Result of Model 0 Regression (per 1 million yen, Log Price Difference)
log Estimate Std. Error t value Pr(>|t|)
A (Intercept) -0.000004 0.000026 -0.165 0.869
β 0.000002 0.000000 90.773<2e-16
B (Intercept) 0.000041 0.000048 0.863 0.388
β 0.000016 0.000000 72.634<2e-16
C (Intercept) -0.000045 0.000042 -1.069 0.285
β 0.000026 0.000000 72.569<2e-16
D (Intercept) 0.000081 0.000051 1.608 0.108
β 0.000060 0.000001 66.662<2e-16
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4.3.9 Estimation of Market Impact: Model 1
In the model 1, log difference of price and order value are used for regression. And also, the
OFI is decomposed into original six elements. Model 1 is defined as follows.
∆Pt = α + β1Lb+ β2Cb+ β3Ms+ β4Ls+ β5Cs+ β6Mb (88)
According to this definition, result of the regression model in case of stock A is summa-
rized in Table 23. The numbers are standardized through dividing by 1,000,000.
Table 23: Model 1 - Log Price Difference and Order Value (Stock A)
Estimate Std. Error t value Pr(|t|)
(Intercept) 0.0000197 0.0000529 0.372 0.71
Limit Buy, β1 0.0000026 0.0000001 24.533<2e-16
Cancel Buy,β2 -0.0000026 0.0000001 -19.343<2e-16
Market Sell,β3 -0.0000027 0.0000001 -26.089<2e-16
Limit Sell, β4 -0.0000024 0.0000001 -22.991<2e-16
Cancel Sell,β5 0.0000026 0.0000001 19.743<2e-16
Market Buy,β6 0.0000021 0.0000001 21.778<2e-16
Table 23 shows that signs of estimated coefficients have negative in Cb, Ms, and Ls and
positive in Lb, Cs, and Mb. This result supports result of the model 0. Therefore, signifi-
cance of OFI is statistically proven. The p-value and t-value show that all coefficients except
intercept are statistically significant.
117
For other four stocks, coefficients are summarized in Table 24. Since magnitude of coef-
ficients are different from stock to stock, one figure which includes all four stocks compares
magnitude and the other figure which separately illustrates coefficients compares relative dif-
ference among six elements.
Table 24: Model 1 - Log Price Difference and Order Value (All Stocks, Coefficients)
A B C D
(Intercept) 0.0000197 0.0000240 0.0001650 -0.0001101
Limit Buy, β1 0.0000026 0.0000178 0.0000306 0.0000627
Cancel Buy ,β2 -0.0000026 -0.0000193 -0.0000332 -0.0000685
Market Sell,β3 -0.0000027 -0.0000175 -0.0000320 -0.0000802
Limit Sell, β4 -0.0000024 -0.0000141 -0.0000221 -0.0000470
Cancel Sell,β5 0.0000026 0.0000160 0.0000231 0.0000541
Market Buy,β6 0.0000021 0.0000142 0.0000227 0.0000674
According to the Table 24, impact is the largest in case of stock D. Stock C is the second,
stock B is the third, and stock A is the least. This order is the same with previous results.
Therefore, if liquidity is high, such as stock A and stock B, single order cannot influence
entire market. On the other hand, if liquidity is low such as stock D, the impact is larger.
Trend between market sell and market buy is the same for all stocks. Market sell which
influences downward pressure to price has larger impact than market buy which upwardly
pressure price change.
Inside one side of the book, cancel orders have relatively higher impact than limit and
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market orders except stock D. On the other hand, market orders have almost the same or
slightly higher impact than limit orders. In case of stock D, market orders have the largest
coefficients and limit orders have the least coefficients.
Other results of this regression are summarized in Table 25.
Table 25: Other Results of Model 1 (All Stocks, Coefficients)
A B C D
Residual Std. Error 0.001193 0.002097 0.001966 0.002338
df 2153 1937 2189 2171
F-statistic 1389.0 890.6 910.2 765.5
df 6, 2153 6, 1937 6, 2189 6, 2171
p-value <2.2e-16 <2.2e-16 <2.2e-16 <2.2e-16
R-squared 0.7947 0.7340 0.7139 0.6790
Adj. R-squared 0.7942 0.7331 0.7131 0.6781
AIC -22939.36 -18453.04 -21128.90 -20201.43
BIC -22893.94 -18408.46 -21083.35 -20155.94
By summarizing these results, it could be assumed that if liquidity is low, market impact
of market order is the largest among three order types. On the other hand, if liquidity is high,
all impacts are small and difference among order types becomes not significant.
Before discussing other models, joint tests to check ifβ1 = −β4, β2 = −β5, andβ3 = −β6
are considered. Based on the definition, estimated coefficients have the following character-
119
istics.
β̂ |X ∼ Nk(β, σ2(X′X)−1) (89)
Here, β̂ = [α̂, β̂1, β̂2, β̂3, β̂4, β̂5, β̂6]′ and Var[ϵ |X] = σ2In. The null hypothesis isH0 :
β1 = β4, β2 = β5, β3 = β6 which means limit orders, cancel orders, and market orders
have the same magnitude of impact in both side of the limit order book. By using following
definitions, joint test can be done.
R=
0 1 0 0 1 0 0
0 0 1 0 0 1 0
0 0 0 1 0 0 1
q =
0
0
0
test= (Rβ̂ − q)′[Rσ2(X′X)−1R′](Rβ̂ − q)
a,H0∼ χ2(3) (90)
Using this Wald test method, following other three hypotheses are also tested.
• β1=−β4, β2=−β5, andβ3=−β6 (Lb=-Ls, -Cb=Cs, and -Ms=Mb, Test 1)
• β1=−β2=−β3 and−β4=β5=β6 (Lb=-Cb=-Ms and -Ls=Cs=Mb, Test 2)
• β1=−β2 and−β4=β5 (Lb=-Cb and -Ls=Cs, Test 3)
• β1 = −β3 and−β4 = β6 (Lb=Ms and Ls=Mb, Test 4)
120
Test 1 is verifying whether limit, cancel, and market orders on different side have the
same magnitude of impact. Test 2 is checking if three elements in the same side of the book
have the same size of coefficients. In test 3, limit and cancel orders on the same side are
tested whether they are the same or not. Finally, test 4 observes that limit and market orders
in the same side have same magnitude of impact.
For these four tests, value to be compared with distribution named "test", p-value, and
critical value are described in Table 26.
Table 26: Model 1 - Log Price Difference and Value Test (All Stocks)
Test 1 Test 2 Test 3 Test 4
Critical Value 7.815 9.488 5.991 5.991
A test 19.275 10.850 7.294 7.002
p-value 0.000 0.028 0.026 0.030
B test 10.287 11.187 10.659 0.222
p-value 0.016 0.025 0.005 *0.895
C test 54.082 12.363 8.990 0.942
p-value 0.000 0.015 0.011 *0.624
D test 15.591 42.056 9.844 31.822
p-value 0.001 0.000 0.007 0.000
In the Table 26, p-values which are larger than 0.05 have "*". Although all settings for
test 1, 2, and 3 are rejected, test 4 has some settings whose p-values are large and cannot
reject the hypotheses. In case of stock B and stock C, which are middle in terms of liquidity,
121
test for limit and market orders in the same side of the book cannot be rejected. That means
limit and market orders in the same side cannot conclude to be statistically different.
122
4.3.10 Estimation of Market Impact: Model 2
Although the model 1 explains price change well, it should be checked carefully. As it
was shown in the previous section, correlation among OFI elements are high. Therefore,
multicollinearity could be included in the model 1. In order to check its existence, Variance
Inflation Factor (VIF) is calculated. VIF is defined as follows.
V IFj =1
1− R2j
(91)
Among independent variables, one variable is set to be target dependent variable. Then,
other independent variables are used to estimate the target variable. If one independent vari-
able could be explained using other independent variables, the independent variables are
highly correlated and some of them are not necessary for regression. In the OFI model, one
of Lb, Cb, Ms, Ls, Cs, or Mb is used as a target variable and other five variables are used to
estimate it. After making regression model,R2 is extracted and VIF is calculated using the
definition. Using six months data, VIF is calculated and summarized in Table 27.
Table 27: VIFs of Model 1
Lb Cb Ms Ls Cs Mb
A 7.240 3.844 2.738 5.529 3.326 3.486
B 8.438 4.629 3.442 6.230 3.727 4.255
C 6.816 3.958 2.971 10.721 8.010 3.110
D 5.538 3.136 3.135 4.604 3.092 2.860
According to the Table 27, limit buy and limit sell orders are able to be explained by other
123
variables. In order to solve this problem, some of the OFI elements can be eliminated from
the dataset.
In this model, cancel order is eliminated because of the optimization model setting. In the
optimization model, traders can use limit orders during the trading periods and market orders
in the terminal period. However, cancel orders are not clearly modelled. Although cancel
orders are used during the periods in order to track the best price, it is not explicitly explained
in the model. What is more, cancel orders are highly correlated with limit orders according to
the correlation matrix of sum of order size in the previous section. Actually, limit buy (sell)
orders and cancel buy (sell) orders are the most correlated combinations among independent
variables. This is because trader cancels previous limit order and replace it to the other
price without changing its size. Therefore, cancel and limit orders are a pair. In the new
model, cancel orders are eliminated. Only limit and market orders are used to explain price
difference. If price difference can be explained using limit and market orders, market impact
will be able to be interpreted easily by observing coefficients in the regression result.
Since four stocks are different ranges of price, they cannot be compared directly using
volume. Therefore, target is decided as value. In this analysis, target is set to be 100,000,000
yen. This target is decided through considering one representative institutional investor in
Japan, Government Pension Investment Fund (GPFI). This fund manages public pension of
Japan and is known as one of the largest institutional investors in the world. According to
their portfolio holdings (GPIF [2017]), assets which are large market values are summarized
as in table 28.
124
Table 28: Top 10 Assets of Portfolio Holdings as of March 31, 2017 (Source: GPIF [2017])
Security Name Num of Shares Market Value (JPY)
TOYOTA MOTOR CORP. 195,737,400 1,202,211,930,800
MITSUBISHI UFJ FINANCIAL GROUP, INC. 1,160,220,400 822,246,170,680
NIPPON TELEGRAPH & TELEPHONE CORP. 124,335,700 598,302,434,400
HONDA MOTOR CO., LTD. 158,031,700 533,356,718,700
SUMITOMO MITSUI FINANCIAL GROUP, INC. 129,189,900 532,262,133,000
SOFTBANK GROUP CORP. 65,466,400 516,135,898,600
MIZUHO FINANCIAL GROUP, INC. 2,398,345,900 498,273,747,975
KDDI CORP. 149,160,600 442,559,171,700
SONY CORP. 93,591,300 353,400,482,800
FANUC CORP. 14,978,300 343,839,270,152
Sum 5,842,587,958,807
They hold domestic equities of 2207 companies as of March 31, 2017. Total market value
of those equities is 34,995,551,349,538 yen. Market values of the top 10 equities account for
one seventh of the total equities. Although this portfolio is not balanced, one equity is around
15,856,615,926 yen in market value. Therefore, 100 million yen, which is the target value
for portfolio restruction in a day, is assumed to be appropriate for our model.
For four stocks, relative importance of 100 million yen can be summarized in Table 29. In
125
the table, daily transacted market order value, target value (100 million yen), and percentage
of 100 million yen compared with daily value are described.
Table 29: Relative Importance of 100 Million yen
A B C D
Daily MO Value (yen) 18,120,300,718 3,441,354,213 1,931,790,779 967,024,868
Target Value (yen) 100,000,000 100,000,000 100,000,000 100,000,000
Target/ Daily Value 0.55% 2.91% 5.18% 10.34%
According to the Table 29, stock A is the most liquid asset. For stock A, 100 million yen
is only 0.55 %. On the other hand, stock D is the least transacted asset where 100 million yen
is over 10 % compared with daily transacted market order value.
Then, target value will be transformed into target volume. In order to do transformation,
price of assets should be decided. Here, reference price is average of closing price for six
months. And also, new measure called "unit" is introduced. Target unit is set to be 100.
Therefore, one unit is equal to 1,000,000 yen. These values are summarized in Table 30.
126
Table 30: Relationship among Value, Volume, and Unit
A B C D
Target Value (yen) 100,000,000 100,000,000 100,000,000 100,000,000
Value/ Unit (yen/unit) 1,000,000 1,000,000 1,000,000 1,000,000
Mean Closing Price 378.08 219.67 234.85 348.79
Volume/ Unit (Asset/ unit) 2645.50 4545.45 4255.32 2865.33
Target Unit (unit) 100 100 100 100
For example, in case of stock C, mean closing price is 235 yen. Since one unit is equal to
1,000,000-yen, one unit is equal to 4255.32 volumes. On the other hand, one unit is 2865.33
volumes in case of stock D. Using unit measure, market impact coefficients are able to be
estimated through regression model described below.
∆Pt =Pt − Pt−1
Pt−1(92)
= α + β1Lbt + β2Mst + β3Lst + β4Mbt (93)
Price fluctuation can be measured as log difference, percentage difference, and absolute
difference. Although there are three possible definitions, due to the computational reason,
percentage difference is chosen. In case of log difference, optimization will contain log and
exponential function which require much time for convergence. On the other hand, absolute
price difference is not able to consider price range of four stocks. Therefore, percentage
difference is chosen as price dynamics due to computation and comparison. Cancel buy
127
and sell order elements are removed from the model due to the high correlation with other
independent variables. The four elements of the Order Flow Imbalance are measured as unit.
Result of this regression is summarized in the Table 31.
Table 31: Result of Regression (per Unit, Percentage Price Difference)
A B C D
Limit Buy, β1 0.0000018 0.0000084 0.0000160 0.0000435
(in %) 0.00018% 0.00084% 0.00160% 0.00435%
Market Sell,β2 -0.0000038 -0.0000276 -0.0000540 -0.0001018
(in %) -0.00038% -0.00276% -0.00540% -0.01018%
Limit Sell, β3 -0.0000016 -0.0000087 -0.0000063 -0.0000369
(in %) -0.00016% -0.00087% -0.00063% -0.00369%
Market Buy,β4 0.0000031 0.0000253 0.0000348 0.0000929
(in %) 0.00031% 0.00253% 0.00348% 0.00929%
By looking at the table, percentage difference which is caused by posting one unit of order
is explained. For example, if the investor posts one unit of limit buy order which is equal to
1,000,000 yen and 4255.32 volume, stock price of stock C will change 0.0016 %. In case of
stock A, if one unit of limit buy order is posted which is 1,000,000 yen and 2645.503 volume,
its price will change 0.00018 %.
Other results of regression are summarized in Table 32.
128
Table 32: Other Result of Regression (per Unit, Percentage Price Difference)
A B C D
Res Std. Error 0.0014 0.0024 0.0024 0.0027
df 2155 1939 2191 2173
F-statistic 1285 884.8 684.7 704.2
df 4, 2155 4, 1939 4, 2191 4, 2173
p-value < 2.2e-16 < 2.2e-16 < 2.2e-16 < 2.2e-16
R-squared 0.7047 0.6461 0.5556 0.5645
Adj. R-squared 0.7041 0.6453 0.5547 0.5637
AIC -22157.08 -17897.89 -20166.30 -19533.61
BIC -22123.01 -17864.45 -20132.14 -19499.50
According to the Table 32, the F test which judges null hypothesis declaring all four co-
efficients are equal to zero is rejected. Therefore, four parameters are statistically significant.
From this table, market impact coefficients of limit buy (β1) and market buy orders (β4)
are used as LMI and MMI.
After eliminating cancel orders, VIF is changed. Improvements are characterized in the
Table 33.
129
Table 33: VIF of Model2 - No Cancel Order
Lb Ms Ls Mb
A 2.441 2.310 2.380 2.587
B 2.837 2.755 2.536 3.057
C 2.601 2.008 1.474 2.567
D 2.724 2.623 2.029 2.531
Compared from the last model which includes cancel orders, VIFs are decreased and all
values are around two. Therefore, multicollinearity problem could be solved to some extent.
130
4.3.11 Estimation of Execution Probability
By using this definition, execution probability is calculated in Table 34 and 35. The table
compares new estimation model with the model proposed by Omura et al. [2000].
Table 34: Estimated Execution Probability (New Model)
A B C D
Min. 0.000 0.000 0.000 0.000
1st Qu. 0.000 0.000 0.000 0.126
Median 0.413 0.078 0.148 0.837
Mean 0.499 0.408 0.401 0.599
3rd Qu. 1.000 1.000 1.000 1.000
Max. 1.000 1.000 1.000 1.000
sd 0.457 0.460 0.450 0.429
skewness 0.052 0.404 0.457 -0.313
kurtosis 1.155 1.279 1.353 1.319
131
Table 35: Estimated Execution Probability (Old Model)
A B C D
Min. 0.000 0.000 0.000 0.000
1st Qu. 0.613 0.416 0.450 0.699
Median 0.905 0.810 0.833 0.937
Mean 0.759 0.676 0.694 0.799
3rd Qu. 1.000 1.000 1.000 1.000
Max. 1.000 1.000 1.000 1.000
sd 0.314 0.356 0.348 0.284
skewness -1.259 -0.758 -0.847 -1.566
kurtosis 3.417 2.137 2.310 4.514
Mean and Median for four stocks is summarized in the Figure 33.
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Figure 33: Estimated Execution Probability (All Stocks)
132
In general, new model estimates execution probability lower than previous literature. This
is because new model considers limit orders more than previous literature. Since the model
of the previous literature considers just difference of depth in the beginning and end, single
order flow is not included. On the other hand, new model tracks limit and cancel orders
continuously. Therefore, it can consider all inflows and outflows occurred at the best price.
133
4.3.12 Conclusions
In conclusion, basic parameter settings for four stocks are summarized in Table 36.
Table 36: Basic Estimated Parameter Settings
A B C D
LMI( β1) 0.0000018 0.0000084 0.0000160 0.0000435
MMI ( β4) 0.0000031 0.0000253 0.0000348 0.0000929
X (Target Unit) 100 100 100 100
P0 (Avg. Closing Price) 378.08 219.67 234.85 348.79
σ 0.00263 0.00406 0.00367 0.00413
E[τ] 0.499 0.408 0.401 0.599
sd[τ] 0.457 0.404 0.450 0.429
134
5 Application of the Optimal Strategy Model
In this section, optimal execution strategy under various parameter settings will be discussed.
First, we will illustrate optimal strategies for four assets under basic parameter settings es-
timated in the previous section. After that, some parameters which are used in optimization
will be changed and sensitivity of optimal strategy will be examined. The parameters to be
discussed are confidence interval of CVaRβ, risk aversion coefficientγ, target order value,
number of periodsN, market impact of limit orderLMI , market impact of market order
MMI , volatility of price changeσ, mean of execution probability E[τ], and standard devia-
tion of execution probability sd[τ]. And also, another assumption for execution probability
which is binomial distribution is compared with normal distribution.
5.1 Basic Parameter Settings
In this section, optimal execution strategies of four stocks under basic parameter settings are
discussed.
Under basic parameter setting of stock A, actual posted order unit which is illustrated as
z(i )k in each trading period is summarized in Figure 34.
135
Figure 34: Posted Order Unit of a set of Simulation Paths
In the first period, all target order unit, 100, is posted. From the next periods, all remaining
unexecuted order unit is replaced as reorder. Therefore, each simulation path takes different
order unit. In any case, optimized order unitxk is not used under this parameter setting. From
the next discussion, posted order unit is described as an average.
By looking at four stocks, average posted order units under the basic parameter settings
are summarized in Figure 35. In the horizontal axis, time periods from t1 to t6 and market
order (MO) at maturity are shown. In the vertical axis, on the other hand, shows average
posted order unit.
136
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Figure 35: Average Posted Order Unit of Four Stocks
According to the Figure 35, all four stocks take early execution strategy. In the early exe-
cution strategy, large order is posted in the first period. Then, order size gradually decreases
as time passes. Through this strategy, trader is able to reduce terminal market order volume
which tends to have larger market impact than limit orders. Especially, stock A, B, and C
have this trend. In these three stocks, all target order unit (100) is posted in the first period.
From the second period, all unexecuted orders will be replaced as limit orders.
This strategy is chosen under two situations, one is lower limit order impact (LMI,β1)
than market order impact (MMI,β4), and the other is lower execution probability. First,
when market impact of limit order is far lower than that of market order, strategy becomes
early execution. Since trader would like to reduce execution cost, limit order is preferred
more if difference of market impact between two types of orders is larger. In the estimated
137
parameters, ratio between LMI and MMI can be computed in Table 37.
Table 37: Relationship between LMI and MMI
A B C D
MMI /LMI 1.676 3.002 2.174 2.138
According to the Table 37, market order has around twice larger impact than limit orders.
Due to this difference, optimal strategy prefers limit orders more than market order.
Although LMI is small in all stocks, stock D is slightly smaller first order size in t1. This
might be due to absolute size of market impact. Stock D is the least liquid asset. That means,
compared with other three stocks, one unit of stock has larger impact than others. If all
target volume is posted at once, that could cause larger market impact than other three stocks.
Therefore, the optimal execution strategy might avoid large order volume in any periods.
Second reason for early execution is lower expectation of execution probability. When
posted order cannot be executed, it will cause unexecution risk. Therefore, larger order tends
to be posted in the first period if the expectation of order execution is small. In the estimated
parameter settings, expectations of execution probability are around 0.5. This is relatively
small. Therefore, optimal strategy becomes early execution.
Next, execution costs of four stocks are compared. Objective function value, expected
execution cost, and CVaR of execution cost are summarized in Figure 36.
138
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According to the Figure 36, all three cost values are the largest in case of stock D. On the
other hand, stock A is the least although stock A, B, and C have almost the same magnitude
of cost values. The objective function value is assumed to be influenced by market impact
parameters since execution probability is almost the same for all stocks. Therefore, stock D
whose estimated market impact parameters are the largest among four stocks has the largest
cost. On the contrary, stock A has the smallest market impact and it has the least cost values.
139
5.2 Sensitivity of the Optimal Strategy to Parameter Settings
5.2.1 Relative Percentage of Target Order Value
Next, magnitude of market impact is discussed through changing target order value and vol-
ume. Target stock is fixed to stock A, 8306 which is the most liquid asset. In the basic
parameter setting, target order value was 100 million yen. However, if this target is changed,
market impact per one order unit is also changed. Parameter settings of target values are
illustrated in Table 38.
Table 38: Target Value Settings
Case-0.552% Case-5.519% Case-27.593% Case-55.187%
Daily MO Value 18,120,300,718 18,120,300,718 18,120,300,718 18,120,300,718
Target Value 100,000,000 1,000,000,000 5,000,000,000 10,000,000,000
Target/ Daily 0.552% 5.519% 27.593% 55.187%
Target Volume 264,550.265 2,645,502.646 13,227,513.228 26,455,026.455
Target Unit 100 100 100 100
Volume/ Unit 2,645.503 26,455.026 132,275.132 264,550.265
Value/ Unit 1,000,000 10,000,000 50,000,000 100,000,000
In the Table 38, target value, target volume, relative percentage of target value to daily
transacted market order value (Target/Daily Value), target unit, volume per one unit, and value
per one unit are described. The four parameter settings are named as Case-X. In X, relative
percentage of the target order value compared with total daily transacted market order value
140
is described. For example, Case-5.519% shows result where target value is 5.519% of the
daily transacted market order value.
Based on these four settings, regression models are constructed. As in the previous sec-
tions, elements of the Order Flow Imbalance are measured as unit. Table 39 shows result of
these regression models.
Table 39: Estimated Coefficients under Target Settings
Case-0.552% Case-5.519% Case-27.593% Case-55.187%
LMI( β1) 0.00000184 0.0000184206 0.000092103 0.0001842063
MMI( β4) 0.00000309 0.0000308730 0.000154365 0.0003087302
Under these four parameter settings, optimization is done. First, average of posted order
value is shown. Figure 37 illustrates optimal average posted order value in four parameter
settings.
141
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For all four settings, posted order value is the largest at the first period. After that, posted
order value gradually decreases. Although this trend is the same for all four cases, this
analysis cannot capture relative order size compared with the target. Therefore, average
posted order unit is illustrated. Figure 38 shows optimal posted order unit.
142
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Figure 38: Average Posted Order Unit under Target Volume Settings
According to the Figure 38, strategy is not the same for four settings. Although all cases
take strategy which posts large order in the first period and gradually decreases order size, not
all target order unit is posted if the target becomes large. For example, when the trader tries to
trade 55.187 % of daily transacted market order value, around 50 % of the target is posted in
the first period. In the latter periods such as period five or period six, the posted order unit is
larger than other cases. Therefore, when target order value becomes larger, optimal strategy
splits large order into small pieces in order not to have large market impact in each period.
Through this splitting, trader is able to reduce large market impact and execution cost.
Objective function value, expected execution cost, and CVaR are summarized in Figure
39.
143
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Figure 39: Execution Cost under Target Volume Settings
Clearly, execution cost increases as target volume increases. In conclusion, target order
volume which is decided by the investor subjectively influences execution strategy dramati-
cally.
144
5.2.2 Confidence Interval of CVaRβ
In this section, confidence interval of CVaR,β, is investigated. Definition of CVaR is as
follows. Stock is fixed to A, 8306, and the parameters are set to Case-0.552% and Case-
55.187%.
CVaR[CN+1] = aβ +1
(1− β)I
I∑t=1
u(i ) (94)
aβ −C(i )N+1 + u(i ) ≤ 0 (95)
0 ≤ u(i ) (96)
Execution cost is defined as following equations.
C(i )k = w(i )
k (P(i )k − P0) +C(i )
k−1 (97)
Given confidence intervalβ, CVaR is defined. For example, whenβ = 0.95, average
of execution cost will be calculated using simulation paths which fall in the largest 5 %.aβ
is VaR (Value at Risk). The confidence interval is changed from zero to 0.99. Due to its
definition, β cannot be one. Under these settings, average posted order is summarized in
Figure 40 and 41. Figure 40 shows Case-0.552% and figure 41 shows Case-55.187%. On the
horizontal axis, confidence intervalβ is described. On the vertical axis, average posted order
which is measured as unit is described. Averages of posted limit order volume from period 1
(t1) to period 6 (t6) together with terminal market order (MO) are shown.
145
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Figure 41: Average Posted Order Unit underβ Settings (Case-55.187%)
146
According to the Figure 40 and the Figure 41, initial limit order unit and limit order unit
in later periods have different characteristics. Whenβ is small or almost zero, limit order is
almost equal size. On the other hand, market order size is large in this parameter setting. In
this case, CVaR considers almost all simulation paths. Due to this reason, CVaR is almost
the same with expected execution cost and trader tries to reduce expected execution cost. In
order to reduce it, large order volume is avoided. Therefore, almost the same size is placed in
each period. On the other hand, whenβ becomes larger or close to one, initial order becomes
large and later orders become smaller. The number of simulation paths which are considered
in CVaR becomes smaller in smallβ settings. Therefore, CVaR is prioritized. CVaR can be
reduced by posting large order volume from the first period.
By comparing two parameter settings of target order value, the Case-0.552% cannot ob-
serve clearer strategic difference than the Case-55.187%. The reason of this characteristic is
relative difference between market impact and volatility of price change. When the target is
small, market impact per one unit is smaller than volatility. Therefore, the impact is assumed
to be almost zero. Under this parameter setting, the investor does not have to care market
impact and post all target order from the first period. When this pressure is strong, strategic
difference byβ setting is not clear. However, if market impact is large and significant for
optimal strategy, changes in objective function through strategic change becomes large. In
this case, strategy varies byβ settings.
In order to clarify strategic difference between early execution and equal size execution,
their execution costs are compared. Whenβ is 0.0 and 0.95 under the Case-55.187%, total
execution cost is able to be summarized in Figure 42.
147
Figure 42: Distribution of Execution Cost underβ Settings (Case-55.187%)
Statistical summary of Figure 42 is described in Table 40.
Table 40: Mean and Standard Deviation of Total Execution Cost (Case-55.187%)
Min. 1st Qu. Median Mean 3rd Qu. Max. skewness kurtosis
β=0.0 -260.5 325.7 434.5 436.4 545.9 1204.8 0.10 3.26
β=0.99 -285.6 345.3 442.4 443.0 540.7 1153.6 0.01 3.24
According to the Figure 42 and Table 40, execution cost is dispersed whenβ is small and
optimal strategy is equal size execution. On the other hand, whenβ is large, execution cost is
centered although its average is larger than smallerβ setting. Therefore, whenβ is small and
148
CVaR is almost the same with expected execution cost, average execution cost is reduced and
standard deviation is ignored through equal size execution. Ifβ is larger and simulation paths
which have large execution cost is influential for overall performance, standard deviation of
execution cost is reduced and average is ignored through early execution strategy.
Objective function, expected execution cost, and CVaR are illustrated in Figure 43 and
Figure 44.
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149
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As β becomes larger, CVaR and objective function value become larger. Whenβ is close
to one, a few simulation paths which have huge execution cost is calculated. Therefore, even
though the strategy is not changed, CVaR becomes larger.
150
5.2.3 Risk Aversion Coefficient γ
Next, risk aversion coefficientγ is investigated. In the optimization model, objective function
is defined as following equation.
min E[CN+1] + γ · CVaR[CN+1] (98)
Parameterγ is multiplied to CVaR (Conditional Value at Risk) of execution cost. If
CVaR is prioritized, optimal strategy tries not to have wide range in execution cost. On the
other hand, if expected execution cost is prioritized, average execution cost is relevant for
optimization.
Stock is fixed to stock A. In these parameter settings, optimal execution strategy is com-
puted. Average posted order unit is depicted in Figure 45 and Figure 46.
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151
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Figure 46: Average Posted Order Unit underγ Settings (Case-55.187%)
As γ becomes larger, initial order volume becomes larger. On the other hand, ifγ is close
to zero, order unit in the first couple of periods becomes smaller and almost the same order
volume is placed in each period. Whenγ is small, weight for risk measure CVaR becomes
smaller and CVaR is assumed to be negligible. In such cases, expected execution cost is
prioritized. Under this situation, posting large order volume could cause dispersed execution
cost due to the variability of execution probability. Therefore, through posting smaller order
volume, investor tries to reduce risk of having uncertainty in execution cost.
Under parameter settings ofγ, objective function value, expected execution cost, and
CVaR are illustrated in Figure 47 and Figure 48. Objective function value is shown as blue
bar, "Obj", expected execution cost is shown as red bar, "E[Cost]", and CVaR is shown as
black bar chart. Ifγ = 0, CVaR cannot be minimized and the results are infinity. Therefore,
152
CVaR is not shown in figures whenγ = 0.
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153
5.2.4 Number of PeriodsN
In this section, the number of periods is changed. In the basic setting, time interval is fixed
to 15 minutes. For another parameter settings, number of periods is changed to be 5 minutes
and 10 minutes. Under these two settings, market impact parameters are estimated again as
in Table 41 and Table 42.
Table 41: Market Impact Parameters (Time Interval: 5 minutes)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0000138 0.0000202 0.685 0.494
Lb, β1 0.0000021 0.0000001 34.373<2e-16
Ms, β2 -0.0000039 0.0000001 -47.462<2e-16
Ls, β3 -0.0000020 0.0000001 -31.313<2e-16
Mb, β4 0.0000034 0.0000001 44.871<2e-16
Table 42: Market Impact Parameters (Time Interval: 10 minutes)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0000384 0.0000447 0.859 0.39
Lb, β1 0.0000018 0.0000001 23.367<2e-16
Ms, β2 -0.0000034 0.0000001 -30.853<2e-16
Ls, β3 -0.0000021 0.0000001 -25.909<2e-16
Mb, β4 0.0000036 0.0000001 37.228<2e-16
154
For comparison, the number of periods is set to 18 when interval is 5 minutes, 9 when in-
terval is 10 minutes, and 6 when interval is 15 minutes. Under these three settings, execution
strategies are optimized. Average of unexecuted order unit is summarized in Figure 49.
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According to the Figure 49, optimal strategy tries to divide large target order into small
pieces. Although there are small differences among three settings, general trend is the same
for all cases. Moreover, two settings which are 10 minutes and 15 minutes of intervals are
similar. Market impact parameters are estimated using different intervals. This is assumed to
allow optimal strategies to consider split of total order according to the number of intervals
N. For example, when time interval is short and the number of trading periods is large, mar-
ket impact is relatively large if large order unit is posted. Therefore, optimal strategy divides
total target into small pieces. Whereas if time interval is long and the number of periods is
155
small, market impact is relatively small and large order is able to be placed. Through consid-
ering trade-off relationship between number of periods and market impact, final answers are
assumed to be similar. The number of split is not optimized in this research and this is able
to be discussed in future extensions.
Objective function, expected execution cost, and CVaR are summarized in Figure 50.
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Three cost values are close under three settings. However, under those parameter set-
tings, objective function value and other cost values are the smallest when time interval is 10
minutes. Therefore, optimal number of split is assumed to be 9 in this case.
156
5.2.5 Market Impact of Limit Order LMI
Then, market impact of limit orderLMI will be changed. This parameter is estimated in
the previous section. By fixing market impact of market orderMMI and changingLMI ,
effect of market impact is discussed. Stock is fixed to A, 8306. As in the previous sections,
Case-0.552% and Case-55.187% are used. Average posted order unit under LMI settings are
summarized in Figure 51 and Figure 52.
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Figure 51: Average Posted Order Unit under LMI Settings (Case-0.552%)
157
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Figure 52: Average Posted Order Unit under LMI Settings (Case-55.187%)
According to the Figure 51 and the Figure 52, order unit is large whenLMI is small.
When market impact of limit order is far smaller market impact than that of market order, the
investor has incentive to use limit order rather than market order in order to reduce cost. By
placing large order from the first period, unexecuted order unit can be reduced aggressively
so that market order is not used a lot. AsLMI becomes larger, initial order size becomes
smaller and becomes equal size execution. IfLMI is larger thanMMI , market order size
becomes larger. The investor does not have incentive to use limit order if its impact is almost
the same or larger than market order. Therefore, optimal strategy shifts to market order from
limit order. Under these parameter settings, all target is placed as market order. In conclusion,
relative difference betweenLMI andMMI influences optimal strategy.
Cost values underLMI settings are summarized in Figure 53 and Figure 54.
158
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As LMI becomes larger, all three cost elements become larger. However, ifLMI be-
159
comes larger thanMMI , almost all target is placed as market order. Therefore, changes in
LMI does not affect cost and increase in cost will be stopped in these parameter settings.
160
5.2.6 Market Impact of Market Order MMI
Oppositely, this section discusses market impact of market orderMMI . Average posted order
unit under two target settings are depicted in Figure 55 and Figure 56.
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Figure 55: Average Posted Order Unit under MMI Settings (Case-0.552%)
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Figure 56: Average Posted Order Unit under MMI Settings (Case-55.187%)
161
In case of Case-0.552% strategic difference underMMI settings cannot be observed. This
is because absolute size of market impacts. Since market impact of limit order when Case-
0.552% is almost zero compared with price volatility, posting all target from the first period is
optimal. In this case, market order is not used. Therefore, changes inMMI does not influence
entire strategy. On the other hand, Case-55.187% shows strategic change. The Figure 56
shows that all target order unit is posted as market order ifMMI is small. Compared with
LMI = 0.000184,MMI is prioritized if MMI is smaller thanLMI . As MMI becomes
larger thanLMI , initial limit order size becomes larger. Moreover, order size from period 4
gradually decreases asMMI is far larger thanLMI . If MMI increases more, all target order
unit could be placed in the first period and replace all unexecuted order from the second
period so that market order is not used.
Objective function value, expected execution cost, and its CVaR are summarized in Figure
57 and Figure 58.
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162
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Figure 58: Execution Cost under MMI Settings (Case-55.187%)
As in the previous discussion of average posted order unit, there is no clear changes in
the Case-0.552%. The Case-55.187% shows changes byMMI settings. WhenMMI is close
to zero, all target is executed as market order and expected execution cost is zero. In this
case, CVaR is only caused by unpredictable price fluctuation. However, asMMI becomes
larger, three cost elements become larger. Although they increase, costs are almost the same
if MMI is larger than 0.0004. Since limit order is preferred and market order is not used,
growth in MMI does not influence cost.
163
5.2.7 Mean of Execution ProbabilityE[τ]
Now, effects of execution probability distribution to optimal strategy is analyzed. Due to its
definition and generation of random numbers, mean cannot be zero or one if the distribution
is truncated normal distribution. Using these parameter settings, average posted order unit is
summarized in Figure 59 and Figure 60.
!
"!
#!
$!
%!
&!!
&"!
!'& !'" !'# !'#(( !'$ !'% !'%) !'( !'(" !'(# !'() !'($
*+,-./,01234,506-5,-07894
:,.802;0<=,>?492801-2@.@9A94B
4& 4" 4C 4# 4) 4$ :6
Figure 59: Average Posted Order Unit under E[τ] Settings (Case-0.552%)
164
!
"!
#!
$!
%!
&!!
&"!
!'& !'" !'# !'#(( !'$ !'% !'()
*+,-./012/.20345-
6.740+809:.;<-5+40*2+=7=5>5-?
-& -" -@ -# -) -$ 61
Figure 60: Average Posted Order Unit under E[τ] Settings (Case-55.187%)
When the mean is close to zero, all target order unit is posted in the first period and all
unexecuted order will be replaced from the second period. In this case, the trader cannot
expect good execution probability and he has to place limit order aggressively in order not
to use market order a lot. On the other hand, if the mean becomes larger and close to one,
initial order unit becomes smaller and posted order unit in each period becomes almost the
same in the Case-55.187%. In the Case-0.552%, posted order unit is also slightly less than
the target. When the mean is large, unpredictability in order execution can be ignored and
large order unit does not have to be posted early. The investor can rather split the target into
smaller pieces. Therefore, the optimal strategy becomes equal execution.
Three cost values are illustrated in Figure 61 and 62.
165
!
"!
#!!
#"!
$!!
$"!
%!!
%"!
&!!
&"!
"!!
!'# !'$ !'& !'&(( !') !'* !'*" !'( !'($ !'(& !'(" !'()
+,-./01/23,45670./890:-:7;76<
=:> 2?@0A6B @C-D
Figure 61: Execution Cost under E[τ] Settings (Case-0.552%)
!
"!!
#!!
$!!
%!!
&!!!
&"!!
&#!!
!'& !'" !'# !'#(( !'$ !'% !'()
*+,-./0123
4305.+6.78392-:+5.;<+=0=:1:->
?=@ 7A*+,-B */0C
Figure 62: Execution Cost under E[τ] Settings (Case-55.187%)
According to the Figure 61 and the Figure 62, cost values are large when the mean is
close to zero. This is because terminal market order volume is large. On the other hand, as
166
the mean becomes larger, cost values decrease since limit order is used in order to reduce
market order size.
167
5.2.8 Standard Deviation of Execution Probability sd[τ]
In this section, standard deviation of execution probability is analyzed. Here, standard de-
viation means standard deviation of original normal distribution. In generation of truncated
normal distribution, normal distribution is used. By giving mean of truncated normal distri-
bution and standard deviation of original normal distribution, random number is made. The
standard deviation is changed from 0.2 to 1.5. Under these parameter settings, Figure 63 and
Figure 64 show average posted order unit.
!
"!
#!
$!
%!
&!!
&"!
!'" !'( !'# !') !'$ !'* !'%
+,-./0-12345-617.6-.189:5
;5/96/.61<-,:/5:3913=1>3.?/@1<:45.:AB5:39
5& 5" 5( 5# 5) 5$ C7
Figure 63: Average Posted Order Unit under sd[τ] Settings (Case-0.552%)
168
!
"!
#!
$!
%!
&!
'!
!(# !(% !(%&) !(' !(* " "(&
+,-./01230/31456.
7.85083019/:68.6,51,;1<=/>?.6,51+3,@[email protected]
." .# .$ .% .& .' C2
Figure 64: Average Posted Order Unit under sd[τ] Settings (Case-55.187%)
Compared with previous parameter settings, optimal strategy is not changed by standard
deviation. Although the first order is the largest and size decreases gradually, the posted order
unit is almost the same under all parameter settings. This is because changes in standard
deviation of normal distribution does not influence standard deviation of generated truncated
normal distribution. Standard deviation of generated truncated normal distribution under
some settings of original normal distribution is described in Figure 65. The horizontal axis
shows standard deviation of original normal distribution and the vertical axis shows standard
deviation of generated truncated normal distribution. Range of standard deviation of original
normal distribution is from 0.05 to 2.0.
169
Figure 65: Standard Deviation of ND and TND
The Figure 65 shows that wide range of original standard deviation of normal distribu-
tion is not resulted in various standard deviation of truncated normal distribution settings.
Generated truncated normal distribution has standard deviation only from 0.1 to 0.3. Com-
pared with original normal distribution, this range is not wide. Therefore, changes in normal
distribution does not affect generated truncated normal distribution and the actual strategy is
almost the same for all parameter settings.
Three cost values are summarized in Figure 66 and Figure 67.
170
!
"!
#!!
#"!
$!!
$"!
%!!
!&$ !&% !&' !&" !&( !&) !&*
+,-./-0/12345-,56.16718609-:125;,05<=,56.
><? @AB6;,C BD-E
Figure 66: Execution Cost under sd[τ] Settings (Case-0.552%)
171
!
"!!
#!!
$!!
%!!
&!!
'!!
(!!
)!!
*!!
!+# !+% !+%&( !+' !+) " "+&
,-./012345
6/2782980:5;<2/<-70-=0>?5@4/<-70A9-B2B<3</C
DBE >F,-./G ,12H
Figure 67: Execution Cost under sd[τ] Settings (Case-55.182%)
Since optimal strategy is almost the same for all parameter settings, cost values are not
changed. However, three cost elements become slightly larger as the standard deviation be-
comes larger. This is because of variability in execution probability. If the execution proba-
bility is varied by simulation paths, terminal execution cost could also be varied. Therefore,
CVaR and objective function value increase more.
172
5.2.9 Volatility of Price Changeσ
Finally, volatility of price changeσ is changed. Under some parameter settings, average
posted order unit is described in Figure 68 and Figure 69.
!
"!
#!
$!
%!
&!!
&"!
! !'!!!!& !'!!!& !'!!& !'!!"$(
)*+,-.+/0123+4/5,4+,/6783
28.9-
3& 3" 3( 3# 3: 3$ ;5
Figure 68: Average Posted Order Unit underσ Settings (Case-0.552%)
173
!
"!
#!
$!
%!
&!
'!
(!
)!
! !*!!!!" !*!!!" !*!!" !*!!#'$ !*!"
+,-./01230/31456.
-6789
." .# .$ .% .& .' :2
Figure 69: Average Posted Order Unit underσ Settings (Case-55.187%)
According to the Figure 68 and the Figure 69, posted order unit in the first period is
smaller than other parameter settings whenσ is close to zero. In the Case-55.187%, average
posted order unit in each period is almost the same size under these parameter settings. As
σ becomes larger, initial order unit becomes larger and later periods do not post large order.
This is able to be explained by unpredictability in price. Although market impact is signifi-
cant when the target is over 50% of daily market order value, volatility of price change still
influences the strategy. When the volatility is large, initial order unit becomes larger in order
to reduce timing risk through having unexecuted order until later periods. Through aggres-
sive reduction of unexecuted order, timing risk is able to be reduced. On the other hand, if
the volatility is small, unpredictability of price change can be ignored. Therefore, optimal
174
strategy posts moderate size in the first period.
Objective function value, expected execution cost, and CVaR are summarized in Figure
70 and Figure 71.
!
"!
#!!
#"!
$!!
$"!
%!!
! !&!!!!# !&!!!# !&!!# !&!!$'%
()*+,
-./ 0123(45 26,7
Figure 70: Execution Cost underσ Settings (Case-0.552%)
175
!
"!!
#!!
$!!
%!!
&!!!
&"!!
&#!!
&$!!
! !'!!!!& !'!!!& !'!!& !'!!"$( !'!&
)*+,-./012
+345/
678 9:)*+,; )./<
Figure 71: Execution Cost underσ Settings (Case-55.187%)
The figure 70 and the figure 71 show that objective function value and CVaR increase as
σ increases. This is because unpredictability in price change exists in largerσ settings. If the
stock price fluctuates aggressively, there is risk which a few conditions have huge execution
cost due to the price increase. Therefore, CVaR becomes larger. However, expected execution
cost is almost the same for allσ settings. Although volatility is large, expectation of standard
deviation is always zero. Therefore, expected execution cost is also constant for variousσ
settings.
176
5.3 Another Model Settings
5.3.1 Binomial Distribution
In this section, distribution of execution probability is changed. In the previous analysis,
truncated normal distribution is used for execution probability. However, many previous
literatures use binomial distribution for execution probability. Here, two distributions are
compared under the same expectation of execution probability. In order to highlight strategic
difference of two distribution, the Case-55.187% is used as a basic parameter setting. If the
Case-0.552% is implemented, all target volume is placed in the first period and all unexe-
cuted order volume is placed from the second period. Under this strategy, changes under
two strategical settings cannot be observed. Due to this characteristic, the Case-55.187% is
considered. Average posted order unit is summarized in Figure 72.
!
"!
#!
$!
%!
&!
'!
(" (# ($ (% (& (' )*
+,-./0-1234(-51*.5-.1678(
9873:8/; <.=7>/(-51?3.:/;
Figure 72: Average Posted Order Unit under Two Distributions (Case-55.187%)
177
Under assumption of binomial distribution, the first, second, and third order is larger than
truncated normal distribution setting. In latter periods, such as the fourth, fifth, and sixth
periods, average posted order unit is larger in case of truncated normal distribution. As a
result, binomial distribution reduces remaining unexecuted order volume more aggressively
than truncated normal distribution.
This strategic difference is able to be explained by standard deviation of execution prob-
ability. In case of binomial distribution, execution probability takes only zero or one. The
probability of appearing one is given by the parameterp. Mean of binomial distribution isp.
Standard deviation of binomial distribution, on the other hand, is given by below definition.
µ = p (99)
σ =√
(p(1− p)) (100)
In case of truncated normal distribution, the value can take continuous value from the
lower bound to the upper bound. In this research, lower bound of execution probability is set
to be zero and upper bound of execution is set to be one. In order to generate random number
of this distribution, inverse transform sampling is used.
First, since mean of the truncated normal distributionµ2 is different from mean of original
normal distributionµ, µ is calculated givenµ2. Median of the distribution is decided through
the following equation.ϕ is probability density function of standard normal distribution and
Φ is cumulative distribution function of standard normal distribution.b is upper bound anda
is lower bound of truncated normal distribution. In case of execution probability,b = 1.0 and
a = 0.0. Meanµ2 and standard deviationσ2 of generated normal distribution is given by the
178
following formulas.
B =b− µσ
(101)
A =a − µσ
(102)
µ2 = µ +ϕ(A) − ϕ(B)Φ(B) − Φ(A)
· σ (103)
σ2 =
√σ2[1 +
Aϕ(A) − bϕ(B)Φ(B) − Φ(A)
− (ϕ(A) − ϕ(B)Φ(B) − Φ(A)
)2] (104)
By minimizing difference betweenµ2 and given target mean of execution probabilityµ,
the mean is decided. Then, random number of truncated normal distributionx is obtained
using following inverse sampling method. In the formula,s is a random number obtained
from unified distribution whose range is from zero to one.
∆ = Φ(B) − Φ(A) (105)
x = σ · Φ−1(∆ · s+ Φ(A)) + µ (106)
Using these definitions, relationship between standard deviation of binomial and trun-
cated normal distribution is illustrated in Figure 73. The figure is drawn by setting given
mean of execution probability for two distributions. Mean of execution probability is shown
in horizontal axis and vertical axis shows standard deviation.
179
Figure 73: Standard Deviation of Binomial and TN Distribution
The black line is binomial distribution and other lines are TND. For TND, there are three
settings of standard deviation of original normal distribution. Green line shows standard
deviation equal to 1.6, red is 0.5, and blue is 0.2. Due to computation limit, standard deviation
cannot be defined in some cases of mean in TND. For example, when standard deviation of
original normal distribution is 1.6, mean of TND can only be in range from 0.4 to 0.8. When
mean of execution probability is outside of this range, standard deviation of TND cannot be
calculated. More precisely, mean of original normal distribution cannot be obtained given
standard deviation of normal distribution and mean of TND.
Under all assumptions of mean value, standard deviation is larger in binomial distribution
although mean is aligned. For example, under basic parameter settings, binomial distribution
has 0.4 standard deviation and TND has around 0.2 standard deviation. Therefore, binomial
distribution is more volatile than TND in definition. It is natural because binomial distribution
can take only two values, zero or one, whereas TND can take continuous value between zero
180
and one. This variability concludes uncertainty of execution. For this uncertainty, order
volume becomes larger in the first periods in order not to hold large unexecuted order volume
in the later periods.
Objective function, expected execution cost, and its CVaR are summarized in Figure 74.
!
"!!
#!!
$!!
%!!
&!!!
&"!!
'() *+,-./0 ,123
456-7528 9:;6<2/=>?@-:728
Figure 74: Execution Cost of Two Distributions
According to this figure, all three cost elements are larger in binomial distribution. Since
unexecuted order volume has larger variability than truncated normal distribution, binomial
distribution tends to make larger execution costs.
181
5.3.2 Market Order Strategy
In order to highlight novelties of this research, some other strategies are compared. In this
section, market order strategy is compared. In this strategy, limit order is not used. The
basic parameter setting is the Case-0.552%. Figure 75 shows average posted order unit for
all periods. One strategy is using limit and market order which is proposed in the previous
sections. The second strategy is using only market order. Market impact of market order is
0.0000018 which is equal to the market impact of limit order of the first strategy. The last
strategy also uses only market order. Here, market impact is fixed to 0.0000031 which is
market impact of market order in the first strategy.
!
"!
#!
$!
%!
&!!
&"!
'& '" '( '# ') '$ *+
,-./01.2345'.62+/6./2789'
:+2;2*+ +<:=*+>**?@!A!!!!!&%B
!
"!
#!
$!
%!
&!!
&"!
'& '" '( '# ') '$ *+
,-./01.2345'.62+/6./2789'
:+2;2*+
+<:=*+>**?@!A!!!!!&%B
+<:=*+>**?@!A!!!!!(&B
Figure 75: Average Posted Order Unit of Market Order and Limit Order Strategy
According to the Figure 75, posted order unit in the first period is not all of the target if
only market order is used. This is because of unexecution risk. When using limit orders, the
182
posted order could not be executed. And also, since market order has larger market impact
than limit order, market order is avoided and large order is posted in the first period. However,
if using market order, this risk does not exist. Therefore, large order is not necessary in
the first period. However, two market impact settings,MMI = 0.000018 andMMI =
0.0000031, are still small compared with price volatility. Therefore, large order is still placed
in the first period.
Execution cost values of these three strategies are summarized in the Figure 76.
!
"!
#!!
#"!
$!!
$"!
%!!
&'( )*+,-./ +012
3&4546&
&7386&966:;!<!!!!!#=>
&7386&966:;!<!!!!!%#>
Figure 76: Execution Cost of Market Order Strategy
According to the Figure 76, expected execution cost is the least when using both limit and
market orders. However, CVaR is the largest when using both limit and market orders. This
can also be explained by unexecution risk. Since limit order has unexecution risk, variability
in terminal market order size is assumed to be large. However, when using market orders,
there is no variability in executed order volume. Therefore, CVaR is less in market order
183
strategy.
184
5.3.3 Target of Market Impact
In this section, target of market impact is changed. In the basic model, market impact is
targeted to executed order volume. This can be modified to posted order volume. Definition
of market impact is changed as follows.
p(i )1 = p0 + p0 · (LMI1 · z(i )
1 + σξ(i )1 ) (107)
p(i )k = p(i )
k−1 + p(i )k−1 · (LMI · z(i )
k + σξ(i )k ) (108)
p(i )N+1 = p(i )
N + p(i )N · (MMI · w(i )
N+1 + σξ(i )N+1) (109)
Using these definitions, posted order volume is optimized in Figure 77.
!
"
#!
#"
$!
$"
%!
%"
&!
&"
"!
'# '$ '% '& '" '( )*
+,-./0-1234'-51*.5-.1678'
9:-;<'-51=3><?- 234'-51=3><?-
Figure 77: Average Posted Order under Two Settings of Market Impact
According to this figure, average posted order is almost the same for all periods when
market impact is targeted to posted order volume. Total execution cost is able to be reduced
185
if posted order volume is equal size in each period. If posted order is larger than other periods
in a few periods, execution cost in the periods will be large no matter the order is executed or
not. Therefore, by posting equal size in each period, total execution cost can be reduced.
186
5.3.4 Strategy without Reorder
Finally, reorder is discussed. In the proposed model, limit order is able to be cancelled if it
is not executed. In order to highlight this novelty, cancel is not allowed. If the posted limit
order cannot be executed, all unexecuted order will be placed as market order in maturity. In
this strategy, the strategy of choosing all unexecuted order volume or optimized order volume
is not implemented. Therefore, optimal strategy is not path dependent. Average posted order
unit is summarized in Figure 78.
!
"!
#!
$!
%!
&!!
&"!
'& '" '( '# ') '$ *+
,-./01.2345'.62+/6./2789'
:;<=>+ :;<=+7<>+
Figure 78: Average Posted Order Unit of Market Order and Limit Order Strategy
According to this figure, posted order unit in the first period is smaller if cancel order is
not allowed. In order to reduce terminal market order, the trader cares total unexecuted order
volume if all unexecuted order becomes market order. Therefore, by reducing order size in
each period, total unexecuted order volume is reduced.
187
Execution cost when not using reorder is depicted in Figure 79.
!
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#!!
#"!
$!!
$"!
%!!
%"!
&!!
&"!
'() *+,-./0 ,123
4567,' 4567'86,'
Figure 79: Execution Cost of Reorder and Without Reorder Strategies
According to this figure, CVaR is far larger without cancel order. Since unexecuted order
volume has large deviation based on execution probability, terminal market order volume
also has wide range if the unexecuted order volume is not replaced as limit orders during
the trading periods. Therefore, CVaR tends to be large in this strategy. In conclusion, using
reorder and reducing unexecuted order volume aggressively could reduce objective function
effectively.
188
6 Conclusions
In this research, optimal limit order strategy for institutional investor is discussed. Replace-
ment of unexecuted order volume is considered in the model. This strategy is able to deal
with unexecution risk which is a typical risk for limit order. Using reorder, execution cost
is able to be reduced. And also, the model is able to consider execution cost more precisely
through introducing market impact for both limit and market orders.
Besides optimization model, some parameters used in the model is estimated using tick
by tick data from the Tokyo Stock Exchange. To be precise, market impact of limit and
market orders, execution probability, and volatility of price change are estimated. Under the
estimation settings, market impacts of limit and market order are not significantly large that
optimal solution does not care this cost. However, if changing some parameters such as risk
aversion coefficient or target value, market impact coefficients become critical for the strategy
and the target is separated into smaller pieces.
In this literature, however, there are some problems which should be considered in the
future extensions. The first problem is execution probability. In the optimization model,
execution probability is independent to the order size. This assumption should be carefully
discussed. Execution probability could be dependent to posted order volume. For example,
if the order size is large, execution probability is assumed to be smaller, whereas if the order
size is very small, it can easily be executed. Therefore, execution probability can be modified
to order size dependent.
The second problem is market impact function. In this literature, price change is assumed
189
to be linear to order size. Although the result shows relatively good fitting, this assumption
can be changed. If the order size is relatively small, the order may not influence the entire
market if its liquidity is good. On the other hand, if the order size is larger than some levels,
its impact can be exponentially large. Due to these reasons, market impact can be nonlinear
to order size.
On the contrary, order pattern can be considered. If the target is separated into small
pieces and post them continuously, the model becomes close to the high frequency trading.
In this case, typical pattern such as order size or frequency can be observed by other traders
and they could use this opportunity.
Finally, the data used for estimation should be updated. The data is obtained in 2012
which is relatively old. Since high frequency trader is critical for the market these days,
using new data such as in 2016 or 2017 could bring other results for this model.
190
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