Master Thesis
Academic Year: 2016-2018
A Study on Optimal Execution Strategy
Model with Limit Orders Using
Multi-Period Stochastic Programming
Involving Unexecution Risk
Shumpei Sakurai
Student 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
3.7 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 of xt . . . 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
4.3 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 CVaR β . . . 145
5.2.3 Risk Aversion Coefficient γ . . . 151
5.2.4 Number of Periods N . . . 154
5.2.6 Market Impact of Market Order M M I . . . 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
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
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 under β Settings (Case-0.552%) . . . 146
41 Average Posted Order Unit under β Settings (Case-55.187%) . . . 146
43 Execution Cost under β Settings (Case-0.552%) . . . 149
44 Execution Cost under β 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 under L M I Settings (Case-0.552%) . . . 159
54 Execution Cost under L M I 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
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
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
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
41 Market Impact Parameters (Time Interval: 5 minutes) . . . 154
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
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 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
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.
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.
!"#$$ !$#%$ $#$$ $#%$ "#$$ "#%$ &#$$ &#%$ '#$$ '#%$ ( ) *! $ % + , -!$ % ./ 0! $ 1 ( ) *! $ 2 + , -!$ 2 ./ 0! $ 3 ( ) *! $ 4 + , -!$ 4 ./ 0! " $ ( ) *! " " + , -!" " ./ 0! " & ( ) *! " ' + , -!" ' ./ 0! " 5 ( ) *! " % + , -!" % ./ 0! " 1 ( ) *! " 2 + , -!" 2 6 7 8 , 9: ;< = ) >: 0: ;? <@ ) ;7 <, A< B C D
Figure 1: Dynamics of Deposit Facility Rate in ECB
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 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
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
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
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
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
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
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
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.
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
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
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
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
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 - ***
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.
mar-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,
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
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 volume y is separated into small pieces in order to trade
within given number of interval N 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 as θ 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)
optimize the model. ytis 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) = min zt
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) V1∗(P0, y1) = y1(P0+ N + 1 2N θy1) (12) = P0X + θy 2 2 (1+ 1 N) (13)
This result shows that optimal order size in each period is the same. The volume is
decided by splitting the total target volume y 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 parameter Mt 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
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 by y units and
this is required to be executed within given maturity T . The number of trading periods is
given by N . Trader posts fractions of target order volume every t = NT 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
as zk. Pk = Pk−1+ σ √ tξk − τg( zk t ) (16) ˜ Pk = Pk−1− h( zk t ) (17)
Posted order volume can be written as zk = 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 term h(·) 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) = ∑t ykg( zk t )+ ∑ zkh( zk t )+ γσ 2∑t y2 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
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
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 as ϵ. After hitting the price, price will be
changed as follows.
Pt− = ¯P (21)
Pt = (1 − ϵ) ¯P (22)
Through minimizing execution cost under iceberg strategy settings, the research discusses
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 is ϕ0, peak of the limit order is shown as ϕp, 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 as ϕs. 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 price n∗ can be written as follows.
n∗= ⌈((ϕ0/ϕp)ϕa+ ϕ0)/ϕs⌉ (23)
The objective function is defined as execution cost.
max ϕ0P¯ (24)
s.t. P∗ ≤ P[n∗ ≤ M] (25)
P0< ¯P (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,
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. Concon-sidering price
distance from the best price is a typical framework for the High Frequency Trading strategy.
In this model, the target order volume y 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 as ˆPtn is
written as follows.
ˆ
Ptn = Ptn + δn (28)
δ shows price distance from the best price at the moment Ptn. Based on the order size and
price distance, execution probability Λ is defined as follows.
Λ(z, δ) = A · exp(−kz − hδ) (29)
k and h 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
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
Ijzj (30)
Ij is a parameter for execution. If it is one, the order zj 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
zkIk(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 order l is used and
cost for using market order can be written as2ly2N−1
fluc-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 2y 2 N−1] (35)
Through backward induction, optimal strategy of zk 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
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
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 cancan-celled 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,
(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.
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
3.7
Objectives, Research Methodology and Research Framework
In this research, main target is an institutional investor who would like to trade y units of
single stock before terminal period t=N. In general, the investor either buys or sells y units or
stock. Here, the investor tries to buy y units of stock in the given trading period. The model
decides optimal limit buy order size in each period t = 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 after N number
of limit buy orders, they will be executed by market order at maturity. Through this strategy,
total traded volume could achieve y 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 coefficient γ.
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)
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
• M MI: 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 Interval β
• u(i): Expected Loss of i-th Path
• w(i)
• 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
min
x1,···,xN
E[CN+1]+ γ · CVaR[CN+1] (36)
s.t. wk(i) = τk(i)z(i)k (37)
z(i)k = min(y(i)k−1, xk) (38)
z(i)N+1 = wN(i)+1 = y(i)N (39)
y(i)k = y(i)k−1− wk(i) (k = 1, · · · , N + 1) (40)
Pk(i) = Pk(i)−1+ Pk(i)−1· (LMI · wk(i)+ σξk(i)) (k = 1, · · · , N) (41)
PN(i)+1 = PN(i)+ PN(i) · (M MI · w(i)N+1+ σξN(i)+1) (42)
Ck(i) = wk(i)(Pk(i)− P0)+ Ck(i)−1 (k = 1, · · · , N + 1) (43)
CVaR[CN+1]= aβ+ 1 (1− β)I I ∑ i=1 u(i) (44) aβ− CN(i)+1+ u(i) ≤ 0 (45) 0 ≤ u(i) (46) N+1 ∑ k=1 w(i)k = y (47)
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 numnum-ber 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
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
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 coefficient γ. Limit order has
unexecu-tion risk and unexecuted orders could influence distribuunexecu-tion of final execuunexecu-tion 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.
min
x1,···,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 interval β. u(i) shows excess amount of
cumulative cost which exceeds VaR aβ.
CVaR[CN+1] = aβ+ 1 (1− β)I I ∑ i=1 u(i) (50) aβ− CN(i)+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.
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
CN(i)+1 100 300 500 600 650 700 720 730 750 800
aβ− CN(i)+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 cost CN(i)+1larger 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 cost u(i). The table 4 shows that u(i) is 20 and
