Proceedings of AESCS2013, Tokyo, Japan

Simulation Study on Eﬀects of Tick SizeDiﬀerence in Stock Markets Competition
Takanobu Mizuta1,2 , Satoshi Hayakawa3 , Kiyoshi Izumi2,4 ,and Shinobu Yoshimura2
2

1
SPARX Asset Management Co. Ltd.
School of Engineering, The University of Tokyo
3
Tokyo Stock Exchange, Inc.
4
CREST & PRESTO, JST

Abstract. We investigated competition, in terms of taking market shareof trading volume, between two artiﬁcial ﬁnancial markets that haveexactly the same speciﬁcations except tick size, i.e., the minimum unitsof a price change per market price, and initial trading volume using multiagent simulations. We found that market share of the trading volume ofa market that adopts a tick size larger than σt (standard deviation of tickby tick return when tick size is small enough) is taken by another market
that adopts a tick size smaller, and market share of the trading volume
of a market that adopts a tick size smaller than approximately 1/10σt
is rarely taken by another market. We also found that agent allocation
rules of limit orders aﬀect the changing speed of trading volume. We
compared these simulation results with empirical data of the Tokyo Stock
Exchange. We argue that these investigations will encourage discussion
about adequate tick sizes that markets should adopt.
Keywords: Stock Market Competition, Tick Size, Artiﬁcial Market,
Multi Agent Simulation

1

Introduction

Recently, the number of stock markets that make full use of IT and achieve
low-cost operations is increasing, especially in the United States and Europe.
Their market shares of trading volume have caught up with those of traditional
stock exchanges. Therefore, each stock is traded at many stock markets at once.
Whether such fragmentation makes markets more eﬃcient has been debated [1,2]. There are many factors that cause the taking of market share of trading
volume between actual markets, such as the minimum unit of a price change
per market price(tick size), speed of trading systems, length of trading hours,
stability of trading systems, safety of clearing, and variety of order types. It
is said that the smallness of tick size is one of the most important factors tocompete with other markets.
Because many factors result in the taking market share of trading volume of
one actual market by another, an empirical study cannot isolate the relationship- 235 -

Proceedings of AESCS2013, Tokyo, Japanbetween tick size and market share of trading volume. Therefore, it is diﬃcult
to discuss the eﬀect of tick size from only the results of empirical studies. An
artiﬁcial market, which is a type of agent-based simulation, will help us determine eﬀects on institutions and regulations in ﬁnancial markets. Many studies
have investigated the eﬀect on several institutions and regulations using artiﬁcial market simulations [3–5]. No simulation studies, however, were focused oninvestigating the eﬀect of the relationship between tick size and market share of
trading volume on stock markets using an artiﬁcial market model.
In this study, we investigated competition, in terms of taking market shareof trading volume, between two artiﬁcial ﬁnancial markets that have exactly
the same speciﬁcations except of tick size and initial trading volume using an
artiﬁcial market model. We found that market share of the trading volume of amarket that adopts a tick size larger than σt (standard deviation of tick by tickreturn when tick size is enough small) is taken by another market that adopts
a tick size smaller than that of the market, and market share of the trading
volume of a market that adopts a tick size smaller than approximately 1/10σt
is rarely taken by another market. We also found that agent allocation rulesof limit orders aﬀect the changing speed of trading volume. We compared thesesimulation results with empirical data obtained from the Tokyo Stock Exchange.
The paper is structured as follows. In section 2, we explain details of our artiﬁcial
market model. In section 3, we show simulations results. In section 4, we explain
the results of an empirical study on the Tokyo Stock Exchange data. We conclude
the paper in Section 5.

2

Artiﬁcial Market Model

We built a simple artiﬁcial market model on the basis of the models developedby Mizuta et al. [4] and Chiarella et al. [6]. The model treats only one risk
asset and one non-risk asset (cash) and adopts a continuous double auction1 todetermine the market price of a risk asset. The number of agents is n. First, attime t = 1, agent 1 orders to buy or sell the risk asset; then at t = 2 agent 2
orders to buy or sell. At t = 3, 4, , , n, agents 3, 4, , , n respectively order to buy
or sell. At t = n + 1, going back to the ﬁrst agent, agent 1 orders to buy or sell,
and at t = n + 2, n + 3, , n + n, agents 2, 3, , , , n respectively order to buy or sell,
and this cycle is repeated. Note that t passes even if no deals have occurred.
An agent j determines an order price and buys or sells by the following process.
Agents use a combination of the fundamental value and technical rules to form
expectations on a risk asset return. The expected return of agent j is
1
t
re,j
=∑
i wi,j
1

(
)
Pf
t
t
w1,j log t + w2,j rh,j + w3,j ϵj ,P

(1)A continuous double auction is an auction mechanism where multiple buyers andsellers compete to buy and sell ﬁnancial assets in the market and where transactions
can occur at any time whenever an oﬀer to buy and an oﬀer to sell match [7].

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Proceedings of AESCS2013, Tokyo, Japan

where wi,j is the weight of term i of agent j, which is independently determinedby random variables of uniform distribution inside the interval (0, wi,max ) at thestart of the simulation for each agent, Pf is a fundamental value that is constant,
P t is the market price of the risk asset at time t. (When dealing does not occur
at t, P t remains at the last market price P t−1 , and at t = 1, P t = Pf ), ϵtj is
noise determined by random variables of normal distribution with an average 0tand variance σϵ , rh,j
is a historical price return inside an agent’s time interval τj ,t
and rh,j= log (P t /P t−τj ) in which τj is independently determined by randomvariables of uniform distribution inside the interval (1, τmax ) at the start of thesimulation for each agent. The ﬁrst term of Eq. (1) represents a fundamental
strategy: an agent expects a positive return when the market price is lower than
the fundamental value and vice verse. The second term of Eq. (1) represents atechnical strategy: an agent expects a positive return when the historical market
return is positive and vice verse. After the expected return has been determined,
the expected price isttPe,j= P t exp (re,j).
(2)
t
We modeled an order price Po,jby random variables of normal distribution int
an average Pe,j , a standard deviation Pσ , where Pσ is a constant. A minimum
unit of a price change is ∆P , and we round oﬀ a fraction of less than ∆P . To
buy or sell is determined by the magnitude of the relation between the expectedttprice Pe,jand the order price Po,j
, that is,tt, the agent orders to buy one share,> Po,jWhen Pe,jttWhen Pe,j< Po,j, the agent orders to sell one share.

Agents always order only one share. Our model adopts a continuous double
auction, so when an agent orders to buy (sell), if there is a lower price sell order(a higher price buy order) than the agent’s order, dealing immediately occurs.
We call such an order “market order”. If there is not a lower price sell order (ahigher price buy order) than the agent’s order, the agent’s order remains in theorder book. We call such an order “limit order”. The remaining order is canceled
after tc from the order time. Agents can short sell freely. The quantity of holding
positions is not limited, so agents can take any shares for both long and shortpositions to inﬁnity.
We investigated the situation in which agents can trade one kind of stock in
two stock markets. The two stock markets have exactly the same speciﬁcations
except minimum unit of a price change, ∆PA , ∆PB and initial share of trading
volume, WA , WB . The agents should decide to which market they order, market
A or market B. When the agent order is buy (sell), the agent searches the lowest
sell (highest buy) orders of each market. We call these prices “best prices”. Whenbest prices diﬀer between two markets and the order will be a market order inleast one of the markets, the agent orders to buy (sell) in a market in which thebest price is better, i.e., lower (higher) in the case of the buy (sell) order. In
other cases, i.e., when the best prices are exactly the same or the order will be- 237 -

Proceedings of AESCS2013, Tokyo, Japana limit order in both markets, the agent orders to buy (sell) in market A withprobability WA = TA /(TA + TB ), where TA is the trading volume of market A
within last tAB , calculating span of WA and TB is that of market B. Therefore,
WB = 1 − WA = TB /(TA + TB ) is the probability the agent orders to buy (sell)in market B. Before time reaches tAB , WA remains constant at an initial value.
To summarize, if the market order and best prices diﬀer, agents order to buy(sell) in the market in which the best price is better than in another market. In
other cases, agents order to buy (sell) in markets depending on the market shareof trading volume.

Table 1. Stylized facts for various tick sizes
tick size(%)
0.0001%
23.5%
trade rate
about trading
cancel rate
26.2%
number of trades / 1 day
6,361
standard
0.05%
for 1 tickdeviationsfor 1 day (20000 ticks)
0.59%
kurtosis
1.50
lag
1
0.229
autocorrelation2
0.141
coefficient for3
0.109
square return
4
0.091
5
0.078

0.001%
23.5%
26.2%
6,358
0.05%
0.56%
1.48

0.01%
23.4%
26.3%
6,345
0.05%
0.57%
1.45

0.1%
23.1%
26.6%
6,279
0.06%
0.57%
1.10

1%
22.1%
27.6%
6,081
0.16%
1.15%
1.81

0.228
0.141
0.108
0.091
0.078

0.228
0.141
0.108
0.091
0.078

0.210
0.120
0.090
0.075
0.064

0.025
0.013
0.008
0.006
0.004

100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%

0.01%
0.05%
0.10%
0.20%

0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
475
500

Trading share of Market A

Trading share of Market A for various ⊿PA(tAB=5 days, ⊿PB=0.01%)

days

Fig. 1. Time evolution of market share of trading volume in Market A for various∆PA (∆PB = 0.01%)- 238 -

Proceedings of AESCS2013, Tokyo, Japan

0.0001%0.0005%0.0010%0.0020%

0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
475
500

Trading share of Market A

Trading share of Market A for various ⊿PA(tAB=5 days, ⊿PB=0.01%)
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
days

Fig. 2. Time evolution of market share of trading volume in Market A for various∆PA (∆PB = 0.0001%)
Trading share of Market A for various tAB(⊿PA=0.1%，⊿PB=0.01%)
100%
Trading share of Market A

90%
80%
70%

1 day
5 days10 days
50 days

60%
50%
40%
30%
20%
10%
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
475

0%
days

Fig. 3. Time evolution of market share of trading volume in Market A for varioustAB (∆PA = 0.1%, ∆PB = 0.01%)

3

Simulation Results

In this study, we set n = 1000, w1,max = 1, w2,max = 10, w3,max = 1, τmax =10000, σϵ = 0.06, Pσ = 30, tc = 20000, andPf = 10000. We ran simulations tot = 10000000.
3.1

Verification of Model

Table I lists statistics for various tick sizes in which there is only one market.
All statistics are averages of 100 simulation runs. Tick size is the ratio of theminimum unit of a price change ∆P to the fundamental value Pf . We deﬁne
20000 time steps as 1 day because the number of trades within 20000 time steps
is almost the same as that in actual markets per day. Trades rates and cancel- 239 -

Proceedings of AESCS2013, Tokyo, Japan

Trading share of Market A for various tAB, for all order volumes(⊿PA=0.1%，⊿PB=0.01%)
100%
Trading share of Market A

90%
80%
70%

1 day
5 days10 days
50 days

60%
50%
40%
30%
20%
10%
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
475

0%
days

Fig. 4. Time evolution of market share of trading volume in Market A for varioustAB (∆PA = 0.1%, ∆PB = 0.01%) or all order volumes

rates 2 are similar in actual markets and verify the model. Standard deviations
for one day are also similar to those in actual markets and also verify the model.
We also found that standard deviations increased with a tick size of 1%.
In many previous artiﬁcial market studies, the models were veriﬁed in termsof whether they can be used to explain stylized facts such as fat-tail and volatilityclustering3 . Table I also lists the stylized facts in each case. We used returns forabout 10 seconds (100 time unit intervals) to calculate the statistical values forthe stylized facts4 . In all runs, we found that both kurtosis and autocorrelationcoeﬃcients for square returns with several lags were positive, which means that
all runs replicate stylized facts. However, only when the tick size was 1%, theautocorrelation coeﬃcients for square returns with several lags were almost zero.
We believe the reason is that the standard deviation of returns is almost constant because the tick size is too big and most returns for about 10 seconds will
be zero or 1%.
3.2

Time Evolutions of Market Share of Trading Volume

We investigated the transition of market shares of trading volume involving
two markets. The two stock markets, markets A and B, had exactly the same2

We referred to the statistics of actual markets in [8] and Bloomberg. Trade rate is
the ratio of the number of trades to that of all orders + cancels. Cancel rate is theratio of the number of cancels to that of all orders + cancels
3
Excellent reviews are [9, 10]. Fat-tail means that the kurtosis of price returns is
positive. There have been empirical studies on fat-tail [11, 12]. Volatility-clusteringmeans that the square returns have positive autocorrelation, and the autocorrelation slowly decays as its time separation becomes larger. There have been empiricalstudies on volatility-clustering [13, 12, 14].
4
In this model, time passes by an agent just ordering even if no dealing occurred.
Therefore, the returns for one tick (one time) include many zero returns, which will
bias statistical values. This is the reason we use returns for about 10 seconds.

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Proceedings of AESCS2013, Tokyo, Japan

speciﬁcations except minimum unit of a price change, ∆PA , ∆PB and initial
market share of trading volume, WA , WB . We set tAB = 10000 (5 days). Figure
1 shows the time evolution of market share of trading volume of market A forvarious ∆PA . We set initial WA = 0.9 and ∆PB = 0.01%. We found that when
∆PA was lager, market A took market share of trading volume from market
B faster. Figure 2 shows the case in which ∆PB = 0.0001%, which is 1/100
compared with that in Figure 1, and ∆PA also became 1/100. We found thatmarket B could not take market share in spite of the fact that the ratios of ∆PA
to ∆PB were the same as those in Figure 1. Therefore, competition under too
small of tick sizes does not aﬀect the taking of market share of trading volume.
Figure 3 shows the time evolution of market share of trading volume of market
A for various tAB , where ∆PA = 0.1%and∆PB = 0.01%. We found that whentAB was shorter, market B took market share of trading volume faster. Thismeans that tAB is an important parameter for the speed of taking market shareof trading volume. Figure 4 shows same as Figure 3 when agents order to buy(sell) in markets not depending on market share of trading volume but the shareof all orders. The speed of taking trading volume is slower than that in Figure
3. We found that how agents distribute orders to markets aﬀects this speed.
Table 2. Market share of trading volume of Market A at 500 days for various ∆PAand ∆PB⊿PB
Trading share ofMarket A at 500 days 0.0001% 0.0002% 0.0005% 0.001% 0.002% 0.005%
90%
90%
91%
91%
92%
94%
0.0001%0.0002%
90%
90%
90%
91%
91%
94%
0.0005%
89%
90%
91%
91%
92%
94%
0.001%
89%
89%
90%
90%
92%
94%
0.002%
87%
88%
89%
89%
91%
93%
⊿PA
0.005%
84%
85%
85%
84%
87%
92%
0.01%
75%
76%
76%
77%
78%
83%
0.02%
53%
52%
53%
54%
54%
59%
0.05%
5%
5%
4%
5%
5%
5%
0.1%
0%
0%
0%
0%
0%
0%
0.2%
0%
0%
0%
0%
0%
0%

3.3

0.01%
97%
97%
96%
97%
97%
96%
92%
70%
6%
0%
0%

0.02%
99%
99%
99%
99%
99%
99%
98%
93%
23%
0%
0%

0.05%
100%
100%
100%
100%
100%
100%
100%
100%
93%
0%
0%

0.1%
100%
100%
100%
100%
100%
100%
100%
100%
100%
94%
0%

0.2%
100%
100%
100%
100%
100%
100%
100%
100%
100%
100%
96%

Relationship between Tick Size and Taking Market Share

Next, we investigated the relationship between tick size and taking market shareof trading volume. Table 2 lists the market share of trading volume of market
A at 500 days, WA for various ∆PA and ∆PB where tAB was ﬁxed for 5 days.
Market share of trading volume was averaged over 100 simulation runs. Lightshading denotes 10% ≤ WA < 80%, and dark shading denotes WA < 10%. We
also drew the following three border lines,∆PA ≤ ∆PB (dashed line),∆PA < σt ≃ 0.05% (double line),1∆PA <σt ≃ 0.005% (solid line),10- 241 -

(3)
(4)
(5)

Proceedings of AESCS2013, Tokyo, Japan

Table 3. Market share of trading volume of Market A at 500 days for various ∆PA
and tAB
Trading share ofMarket A at 500 days
0.0001%0.0002%0.0005%0.001%
0.002%⊿PA
0.005%
0.01%
0.02%
0.05%
0.1%
0.2%

1
91%
89%
89%
88%
84%
62%
24%
0%
0%
0%
0%

2
89%
90%
88%
88%
86%
73%
53%
12%
0%
0%
0%

tAB
5
90%
90%
89%
89%
88%
85%
78%
59%
8%
0%
0%

10
90%
90%
90%
90%
89%
87%
85%
77%
44%
10%
0%

20
90%
90%
90%
90%
90%
89%
87%
84%
73%
52%
24%

50
90%
90%
90%
90%
90%
89%
89%
88%
85%
80%
74%

Relationship between ⊿PA，σt and Trading share(⊿PB=0.0001%)
1.00%

100%

60%
0.10%
40%

1.0000%0.05%

0%0.1000%0.005%

0.0010%0.0001%⊿PA

0.0100%20%
0.01%

Trading share ofMarket A at 500
days

σt

80%

σt (left axis)
Trading share of Market A at 500 days (right axis)

Fig. 5. Standard deviation of returns for 1 tick, σt and market share of trading volumeof Market A at 500 days for various ∆PA

where σt is the standard deviation of return for one tick, which was small enough.
We found approximately 0.05% from Table 1. Region Equation (3) satisﬁes theregion above the dashed line in Table 2, and region Equations (4, 5) satisfy theregion above the double or solid lines. The region that at least Equation (3) or(5) satisﬁes is not shading, and market share of trading volume of market A israrely taken. The region that both Equations (3) and (4) do not satisfy, under
the dashed and double lines, is mostly darkened, and market share of trading
volume of market A is rapidly taken. In the region above the solid line, Equation(5), market share of trading volume of market A is rarely taken even if ∆PB
is much smaller than ∆PA , and in the region above the double line, Equation(4), market share of trading volume share is not taken rapidly and does notdepend on ∆PB . This means that when the tick size of market A is smaller thanσt , market share of trading volume of market A is slowly taken, and when thetick size is smaller than 1/10σt , market share of trading volume is rarely taken.

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Proceedings of AESCS2013, Tokyo, Japan

Table 3 lists market share of trading volume of market A at 500 days, WA forvarious ∆PA and tAB where ∆PB = 0.0001%. Equation (3) was never satisﬁed
in all regions. We found that when tAB was larger, the taking of market share of
trading volume of market A is slower; however, in the region above the solid line,
Equation (5), market share of trading volume of market A is rarely taken. This
may suggest that market share of trading volume of markets that adopt ticksizes smaller than approximately 1/10σt are rarely taken by other competing
markets.
Figure 5 shows the standard deviation of return for one tick, σt , and market
share of trading volume of market A at 500 days, WA for various ∆PA where∆PB = 0.0001%. σt and WA are averages of 100 simulation runs. The left verticalaxis and horizontal axis are logarithmic scales. Equation (3) was never satisﬁed
for all ∆PA in Figure 5. The horizontal dotted line is σt = σt , and the two dashedlines are ∆PA = 1/10σt and ∆PA = σt . On left side of ∆PA = σt , σt equals
σt and σt does not depend on ∆PA . This means that the diﬀerence in tick size
does not eﬀect price formations where tick sizes are smaller than approximately
1/10σt . On the other hand, on the right side of ∆PA = σt , ∆PA is larger, σt islarger. This implies that the prices normally ﬂuctuate less than ∆PA ; however,price variation less than ∆PA is not permitted. Therefore, price ﬂuctuations
depend on ∆PA . In this case, market share of trading volume rapidly decease
according to increasing ∆PA . On the left side of ∆PA = 1/10σt , however, theshares are stable. Among ∆PA = 1/10σt ∼ σt , the market share slowly decrease
according to increasing ∆PA .
We summarize discussion so far referring Figure 6. When ∆PA is larger thanapproximately σt (Figure 6 top), if ∆PB is smaller than ∆PA , there is a large
amount of trading in market B inside ∆PA . Therefore, market B takes market
share of trading volume from market A. On the other hand, when ∆PA is smaller
than approximately 1/10σt (Figure 6 bottom), even if ∆PB is very small, priceﬂuctuations cross many widths of ∆PA and suﬃcient price formations are occur
only in market A. Therefore, market B can rarely take market share of trading
volume from market A. Among ∆PA = 1/10σt ∼ σt , market B slowly takes
trading volume from market A.

4

Empirical Analysis

Next, we analyzed empirical data and compared them with the simulation results
shown in Figure 5, using Japanese stock market data. The data period included
all business days in the 2012 calendar year. A number of stocks analyzed is 439,
which were selected by TOPIX 500 5 over the entire index data period, had thesame minimum unit of a price change for every month end and were traded every
business day. Figure 7 shows the standard deviations for 10 seconds of each stock,where σt (triangles) is the averaged standard deviation of return for 10 minutes
5

TOPIX 500 is a free-ﬂoat capitalization-weighted index that is calculated based on
the 500 most liquid and highly market capitalized domestic common stocks listedon the Tokyo stock exchange ﬁrst section [15].

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Proceedings of AESCS2013, Tokyo, Japanσt < ⊿PA
price

unabletrading in
Maket A⊿PA
time
unable trading in Maket A→ many trading in Maket B⇒ trading share moving to Makrt B

1/10σt > ⊿PA
price

⊿PA

needless ofMarket B
time⇒ trading share not moving

Fig. 6. Summary of relationship between tick sizes and standard deviations of returns
for 1 tick

except opening prices for every day and ∆P is market share of the trading volumeof the Proprietary Trading System (PTS) 6 of each stock (circles) with tick sizes
that are minimum unit of a price change divided by averaged prices at the endof every month of the Tokyo Stock Exchange. We used the data from the Tokyo
stock exchange to calculate ∆P and σt . We used Bloomberg data to calculate themarket share or trading volume of the PTS, which is its entire trading volume
divided by those of Japanese traditional stock exchanges and PTS, where PTSs
are Japan next PTS J-Market, X-Market, and Chi-X Japan PTS, and where
Japanese traditional stock exchanges are the Tokyo, Osaka, Nagoya, Fukuoka,
and Sapporo stock exchanges and JASDAQ. The right vertical axis is upside
down to easily compare with Figure 5. The horizontal dotted line is σt = σt , andtwo dashed lines are ∆P = 1/10σt and ∆P = σt .
On the left side of ∆P = σt , σt , which equaled σt and σt , did not muchdepend on ∆P . On the right side of ∆P = σt , ∆P was larger, σt was larger.
These results are similar to those in Figure 5. The market share of trading
volume of PTS deceased along with ∆P for the entire ∆P . This result is similar
among ∆PA = 1/10σt ∼ ∆PA = σt in Figure 5. On the left side of ∆P = 1/10σt ,however, we did not argue that shares of PTS are stable such as Figure 5 because
there were not enough data. We did found that when ∆P was larger, PTS more
6

Electric trading systems outside stock exchanges are called PTSs in Japan. A PTS
is very similar to an Alternative Trading System (ATS) and Electronic Communications Network (ECN) in other countries.

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Proceedings of AESCS2013, Tokyo, Japan

Relationships between ⊿P，σt and Trading share（Empirical data for 2012）

0.01%

12%

⊿P

10.00%

9%

1.00%

0.10%

0.05%

6%

0.10%

1.00%

0.005%

3%

Trading shareof PTS0%

10.00%

0.01%

σt

100.00%σt (left axis)
σt=0.05%
Trading share of PTS (right axis)

Fig. 7. Empirical analysis of relationship between tick sizes, standard deviations ofreturns, and market share of trading volume of PTS

easily took market share of trading volume, and on right side of ∆P = σt , σttended to increase along with ∆P .

5

Conclusion and Future Study

We investigated competition, in terms of taking market share of trading volume
between two artiﬁcial ﬁnancial markets that had exactly the same speciﬁcations
except tick size and initial trading volume using multi-agents simulations. Whenthe tick size of market A, ∆PA , was larger than approximately the standard
deviation of tick by tick return when tick size was small enough, σt (Figure 6top), if the tick size of market B, ∆PB , was smaller than ∆PA , much trading
occurred in market B inside ∆PA . Therefore, market B took market share of
the trading volume from market A. On the other hand, when ∆PA was smaller
than approximately 1/10σt (Figure 6 bottom), even if ∆PB was very small, priceﬂuctuations cross many widths of ∆PA and enough price formations occurred
only in market A. Therefore, market B could rarely take market share of trading
volume from market A. Among ∆PA = 1/10σt ∼ σt , market B slowly took market share of trading-volume from market A. We also compared these simulation
results with empirical data from the Tokyo Stock Exchange. We argued that thisinvestigation will enable discussion about the adequate tick sizes markets should
adopt.
One future study is to investigate the eﬀects of high-frequency traders whose
investment strategies including market maker strategies and arbitrage strategy
because they might play important roles for competition between markets interms of market share of trading volume. Another future study is to discuss
other methods with which agents allocate orders to markets because, for exam-- 245 -

Proceedings of AESCS2013, Tokyo, Japanple, investors may allocate orders to a market in which the order book has ashorter queue. The order allocation method in this study might be too simple
for discussing actual markets.

Disclaimer
It should be noted that the opinions contained herein are solely those of the authorsand do not necessarily reﬂect those of SPARX Asset Management Co., Ltd. and Tokyo
Stock Exchange, Inc.

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