8th International Congress on
Advanced Applied Informatics (AAI 2019)http://www.iaiai.org/conference/aai2019/

Agent-Based Model of
Liquidity and Arbitrage Cost
Between ETF and Stocks
Takanobu Mizuta

SPARX Asset Management Co., Ltd.
Mail: mizutata[at]gmail.com
HP: http://mizutatakanobu.com

Note that the opinions contained herein are solely those of the authors anddo not necessarily reflect those of SPARX Asset Management Co., Ltd. andNomura Research Institute, Ltd.

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

(1) Introduction(2) Artificial Market Model(3) Simulation Result(4) Summary

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

(1) Introduction(2) Artificial Market Model(3) Simulation Result(4) Summary

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

ETF and Liquidity
Exchange-Traded Fund (ETF) a mutual fund that invests in a diversified portfolioof many stocks or bonds
 listed and traded at a stock exchange
ETFs have been widely spread to individual investors
as an easy way to diversify their investments.
On the other hand,Some ETFs have not been traded with enough volume(low liquidity) to discover an adequate price, making
them difficult for individual investors to trade.
You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

Exchangeable, Arbitrage Trades
arbitrage trader

buy cheapersell more expensive

ETF

profit fromprice difference

Stock
１

Stock
２

exchangeable

Stock
3

Stock
4

・・・

An ETF is exchangeable with all stocks held by the ETF.
When the price of the ETF and the total value of the stocks held by the
ETF differ, a trader can buy the cheaper asset, exchange, sell the more
expensive asset, and thus earn a profit from the price difference.

Increasing arbitrage traders is increasing liquidity

Market-making incentive Scheme

SupplyLiquidity

Designated market makeralways place orders in return for incentives such as lower fees
MM places(Waiting)
MarketLimit
MakingOrdersETF

MM places(Taking)
Arbitrage
OrdersETF is exchanged forstocks to arbitrage

Stock
１

Stock
2

Stock
3

Stock
4

To increase the liquidity of low-liquidity ETFs, in 2018 the Tokyo
Stock Exchange introduced a market-making incentive scheme,in which designated market makers always place orders inreturn for incentives such as lower fees [JPX 17].
The questions, however, of how liquidity changes depending on arbitrage
trading costs and of what the mechanism is remain to be answered.

Difficulty of Empirical Study
Empirical Studies
 cannot be conducted to investigate situations that havenever occurred in actual financial markets
 cannot be conducted to isolate the direct effect on liquidity
because so many factors affect price formation and liquidity
in actual markets

Therefore, in this study,
I expanded the artificial market model of [Mizuta 13] to include
three risk assets, two stocks and an ETF. I also added an arbitrage
agent to perform arbitrage trading among these risk assets.
I then investigated the relationship between the liquidity of an ETF
and the trading costs.

Artificial Market Simulation (Agent-Based Model)
Virtual and Artificial financial Market built on Computers
ModelsIncludeAgents (Artificial Investors)＋
Price Mechanism (Artificial Exchange)
Order

Price
Mechanism(Exchange)

Agent(Investor) Each Agent determines
an order by some rules,
Price Mechanism gather
agents orders anddetermines Market Price
Determination market priceComplete Computer Simulation needing NO Empirical Data can discuss on the mechanism between the micro-macro feedback can be conducted to investigate situations that have neveroccurred in actual financial markets
 can be conducted to isolate the direct effect of changing the cost ofarbitrage tradesPrevious Contributions of Artificial Market Simulations
Many studies have investigated the effects ofseveral changing financial regulations andrules by using artificial market,
Reduction of Tick Size, Up-Tick Rule,
Price Variation Limit, Dark Pool, Frequently Batch Auction,
Contribution of HFTs for share competition among Exchanges,
Suitable Latency of Exchange System, VaR Shock,
Chain Bankruptcy of Banks,
Regulations and Rules to prevent Financial Crush Mizuta (2019) An agent-based model for designing a financial market thatworks well, arXive, https://arxiv.org/abs/1906.06000
 Mizuta (2016) A Brief Review of Recent Artificial Market Simulation Studies
for Financial Market Regulations And/Or Rules, SSRN Working Paper Series
http://ssrn.com/abstract=2710495

NATURE/SCIENCE articles argued Importance of Simulations
 Farmer and Foley (2009) NATURE, Vol. 460, No. 7256, pp. 685-686.
https://www.nature.com/articles/460685a
In today’s high-tech age, one naturally assumes that US President Barack Obama’s
economic team and its international counterparts are using sophisticatedquantitative computer models to guide us out of the current economic crisis. They
are not. There is a better way: agent-based models.
 Battiston et al. (2016) SCIENCE, Vol. 351, Issue 6275, pp. 818-819.
http://science.sciencemag.org/content/351/6275/818
Traditional economic theory could not explain, much less predict, the near collapseof the financial system and its long-lasting effects on the global economy. Since the2008 crisis, there has been increasing interest in using ideas from complexity
theory to make sense of economic and financial markets.
These articles argued that

Traditional economic theory could not explain, muchless predict, the near collapse of the financial system
Agent-Based Model is neededPractical Persons have began to use Agent-Based Modelto solve Urgent Real Problem
Regulators, Central Bankers, Stock Exchanges
JPX Working Papers Series
JPX(parent com of Tokyo Stock Exchange) shows Working Papers,
9 papers of all 31 are Agent-Based Studies
Reduction of Tick Size, Frequently Batch Auction,
Suitable Latency of Exchange System, and so on
https://www.jpx.co.jp/english/corporate/research-study/working-paper/index.html

So many Examples,Working Paper by Bank of Japan
Toshiyuki Sakiyama and Tetsuya Yamada Market Liquidity and Systemic Risk in Government Bond
Markets: A Network Analysis and Agent-Based Model Approach
http://www.imes.boj.or.jp/research/abstracts/english/16-E-13.html
Project by EU
Integrated Macro-Financial Modelling for Robust Policy Design Work Package 7: Bridging agent-based
and dynamic-stochastic-general-equilibrium modelling approaches for building policy-focused macrofinancial models http://www.macfinrobods.eu/research/workpackages/WP7/wp7.html

(1) Introduction(2) Artificial Market Model(3) Simulation Result(4) Summary

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

ETF & Stocks
exchangeable like as
ETF(one share)＝ Stock 1(one share)+ Stock 2(one share)
There are 3 assets,

stock
1

ETF

＋

stock
2

exchange
fundamental price20000

fundamental price fundamental price
10000
10000

Normal Agent(NA) & Arbitrage Agent(AA)AA places

MarketMaking(Waiting)
Limit
OrdersAA (only one)AA places(Taking)
Arbitrage
OrdersETF is exchanged forstocks to arbitrage
Details of AA will explained after slides,
First, I explain about NAs

ETF

stock
1

stock
2

TradesTradesTradesNA(1000)

NA(1000)

NA(1000)

Only for ETF

1/10 orders
for the Stocks
Lower
Liquidity

NAs place orders with 10% probability to investigate for low liquidity

For each risk asset, the model includes 1000 normal agents (NAs)that trade only that risk asset, giving a total of 3000 NAs.

Normal Agent(NA)I used the model of Mizuta (2013), which is based on Chiarella (2002).
The model is satisfied with stylized facts (statistical characteristics
observed in actual financial markets).

Expected Return of each NA

Technical

Pf
1 
t
t 
 w1, j log t  w2, j rh, j  w3, j j r P
i wi, j 

t
e, j

Fundamental

noise

All NAs use this same equation to obtain an expected return,however, because w is different each agents, expected returns aredifferent each agents. This leads heterogeneous (many order prices
are diversified) although the model is simple.

The simplicity of the model is very important.
Models include too many related factors prevent
understanding and discovery of mechanisms affecting
price formation.

Fundamental and Technical Strategies
Fundamental Strategy
Fundamental Price ＞ Market Price -> Expect + return
Fundamental Price < Market Price -> Expect - return
Technical Strategy
Historical Return ＞ 0 -> Expect + return
Historical Return < 0 -> Expect - return
Technical
Strategy

Fundamental
Price
MarketPrice

Fundamental
StrategyDetail of Expected Return
j: agent number (1,000 agents)
ordering in number ordert: tick time

Expected Return of each NA

Historical Return
t
h, j

r

 log( P / Ptt  j)Technical

Pf
1 
t
t 
 w1, j log t  w2, j rh, j  w3, j j r P
i wi, j 

t
e, j

wi , j and  j

Pf

Random ofUniform Distribution

Ptwi , jji=1,3: 0～1
i=2: 0～10
0～10000

noise

Fundamental

Parameters for agents

Fundamental Price
10000 = constantMarket Price at t

Expected Price of each NA tj

Random ofNormalDistribution
Average=0
σ=3%Pet, j  Pt exp(ret, j )Order Price and Buy or Sell
Price

Order Price of each NA

P t o, jSell(one unit)
Gauss
Distribution

Expected Price of each NA

P e, j  P exp( r e, j )tttBuy(one unit)

To replicate many waiting limit orders,
order price is scattered around expected price
NA places one buy order when order price > expected price
NA places one sell order when order price < expected priceArbitrage Agent (AA) (1/4)
Order Frequency
Arbitrage Agent (AA)

t=1

t=2

t=3

t=4

t=5

time

Normal Agent(NA)
The AA can always place orders, change orders, or cancel orders.

Arbitrage Agent (AA) (2/4)
ETF(one share)＝ Stock 1(one share)+ Stock 2(one share)

sell
7
10

ETF
price
20300
20200
20100
20000
19900
19800
19700
19600

buy

10
6
4

sell
30
44
70
134

stock 1
price
10400
10300
10200
10100
10000
9900
9800
9700

buy

120
88
52
25

sell
50
70
90
116

stock 2
price
10400
10300
10200
10100
10000
9900
9800
9700

buy

154
60
55
31

These are for examples of order books.
The sum of the highest buy-order prices for stocks 1 and 2 is 20000 (=10000+100000).
The highest buy-order price for ETF is 19800, and there is no buy order at 19900.
In this case, the AA first places an order to buy one share at 19900 and then waits.

Arbitrage Agent (AA) (3/4)
ETF(one share)＝ Stock 1(one share)+ Stock 2(one share)

sell
7
10

ETF
price
20300
20200
20100
20000
19900
19800
19700
19600

buy

1
10
6
4

sell
30
44
70
134

stock 1
price
10400
10300
10200
10100
10000
9900
9800
9700

buy

sell
50
70
90
116

120
88
52
25

stock 2
price
10400
10300
10200
10100
10000
9900
9800
9700

buy

154
60
55
31

Once the order is matched and the AA buys ETF,
it exchanges the ETF share for stocks 1 and 2
and then sells them each at 10000.
AA earns a profit of 100 from the price difference,
10000 (Stock 1)+ 10000 (Stock 2) – 19900 (ETF) = 100
Of course, the AA can also earn a profit in the opposite case, by first sellingborrowed ETF at a higher price, buying the stocks at lower prices, exchanging thestocks for ETF, and returning the ETF, again earning the price difference as a profit.

Arbitrage Agent (AA) (4/4)
ETF(one share)＝ Stock 1(one share)+ Stock 2(one share)

sell
7
10

ETF
price
20300
20200
20100
20000
19900
19800
19700
19600

buy

1
10
6
4

sell
30
44
70
134

stock 1
price
10400
10300
10200
10100
10000
9900
9800
9700

buy

120
88
52
25

sell
50
70
90
116

stock 2
price
10400
10300
10200
10100
10000
9900
9800
9700

buy

154
60
55
31

When Highest buy-order prices(HBPs) consist
HBP for ETF + Cost ＜ HBP for stocks 1 + HBP for stocks 2
AA first places an order to buy one share at HBP + one tickand then waits.
Note that
cost
includes the required profit, when the price difference ofrisk assets is over c, the AA always do an arbitrage trade.

(1) Introduction(2) Artificial Market Model(3) Simulation Result(4) Summary

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

Market Price Differential rate & Trading Volume of AA
Market price differential rate＝Mid price of ETF/Sum of Mid prices of stocks - 1

0.25%

10000
9000
8000
7000
6000
5000
4000
3000
2000
1000
0

0.20%
0.15%

0.10%
0.05%

No AA

0.500%

0.250%

0.100%

0.050%

0.025%

0.010%

0.005%

0.00%

trading volume of AA

trading volume of AA

0.000%

market price differential ratio

market price differential ratio

cost

Lower cost leads more trading volume and a lower price differential
The price differential sharply changed when the cost was near 0.1%,
similar to the volatility
Whether the cost is higher or lower than the volatility seems toindicate a very important boundary.

Relationship between Cost and Volatility
highest buy-order price of ETF
chance ofarbitrage trade

price

cost

volatility

sum ofhighest buy-order prices oftimestocks 1 and 2

AA can make an arbitrage trade only when the red dashed line is above
the black solid line plus the cost.
Prices of each risk asset fluctuate in their volatility

Volatility > Cost

AA has more chances for arbitrage tradesDepths of Waiting Orders
Sum of waiting orders between Mid Price ± 0.1%

9800

No AA

1000

0.500%

9900

0.250%

1050

0.100%

10000

0.050%

1100

0.025%

10100

0.010%

1150

0.005%

10200

depth of stock 1

stock 11200

0.000%

depth of ETF

ETF

cost

Lower cost leads more depth for ETF and sharply changed when thecost was near 0.1%. On the other hand, the depth for stock 1 had the
opposite tendency.
AA transfers waiting orders (liquidity) from Stocks to ETF

Trading Volume

No AA

0.500%

2700000.250%

27000

0.100%

2800000.050%

28000

0.025%

2900000.010%

29000

0.005%

300000trading volume of stock 1

stock 130000

0.000%

trading volume of ETF

ETF

cost

Lower cost meant higher trading volume for both
Lower cost makes the depth for stock 1 thinner and the trading
volume larger, because orders for arbitrage trades and waitingorders for stock 1 are matched.

(1) Introduction(2) Artificial Market Model(3) Simulation Result(4) Summary

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

Summary
 I expanded the artificial market model of [Mizuta 13] to include
three risk assets, denoted as stock 1, stock 2, and ETF, along withan arbitrage agent (AA) that could perform arbitrage trades
among these risk assets. I then investigated the relationshipbetween the liquidity of ETF and the trading costs.
 My results showed that, because the prices of each risk asset
fluctuate in their volatility, when the volatility is sufficiently greaterthan the cost, the AA has more chances to make arbitrage trades.
As the AA trades more, the market price differential becomes
lower.
 In addition, lower cost means a thicker depth of waiting trades for
ETF, whereas the depth tendency of a stock is the opposite.
Furthermore, lower cost increases the trading volume of both.
Lower cost makes the depth thinner and the trading volume
greater for a stock because the orders for arbitrage trades and thewaiting orders for the stock are matched.

Future Work
 Real financial markets, however, include traders who place moreorders when the trading volume increases. My model did not
implement this behavior. It is possible that lower cost would
increase both the depth and the trading volume with such
behavior. This remains for a future work.

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

That’s ALL, Thanks!!
References-- market-making incentive scheme - [JPX 17] Overview of ETF Market Making Scheme, Japan Exchange Group(JPX),
https://www.jpx.co.jp/english/equities/products/etfs/market-making/-- working paper - [Mizuta 19] Mizuta, T. : Liquidity and Arbitrage Cost between ETF and Stocks using Agent-Based
Model, JPX Working Paper, No. 27, Japan Exchange Group(JPX),
https://www.jpx.co.jp/english/corporate/research-study/working-paper/index.html-- review paper – [Mizuta 19] Mizuta, T.: An agent-based model for designing a financial market that works well,arXive, https://arxiv.org/abs/1906.06000
 [Mizuta 16] Mizuta, T.: A Brief Review of Recent Artificial Market Simulation Studies for Financial
Market Regulations And/Or Rules, SSRN Working Paper Series
http://ssrn.com/abstract=2710495

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

Appendix

You can download this presentation material fromhttp://mizutatakanobu.com/201907.pdf

Stylized Facts

The model of Chiarella (2002) is very simple but replicates long-term statistical characteristics
observed in actual financial markets: a fat tail and volatility clustering.
In contrast, Mizuta (2013) replicates high-frequency micro structures, such as execution rates,
cancel rates, and one-tick volatility, that cannot be replicated with the model of Chiarella (2002).
In this study, I expanded the artificial market model of Mizuta (2013) to include three risk assets,denoted as stock 1, stock 2, and ETF and I added an arbitrage agent to perform arbitrage trading
among these risk assets.
The simplicity of the model is very important for this study, because unnecessary replication ofmacro phenomena leads to models that are overfitted and too complex. Such models prevent
understanding and discovery of mechanisms affecting price formation because of the increase inrelated factors.

Market Inefficiency
Market Inefficiency = Mid Price/Fundamental Price - 1
ETFstock 1market inefficiency

0.20%
0.15%
0.10%

0.05%

No AA

0.500%

0.250%

0.100%

0.050%

0.025%

0.010%

0.005%

0.000%

0.00%

cost

With lower cost, the ETF market became more efficient,but that of stock 1 did not change.
The reason why the ETF market becomes more efficient is
NOT because it gains efficiency from the stock 1 market.

Case with more Liquidity for ETF (Cost=0)(Actually, I fixed the ETF order ratio to k=0.1)

9400

3100009200

3000009000

2900008800

2800008600

270000100%

320000

70%

9600

50%

330000

30%

9800

20%

340000trading volume of stock 1

trading volume

10000

10%

depth of stock 1

depth

ETF order ratio (k)Larger ETF order ratio meant thinner depth and more trading volume.
More ETF orders caused more matching of arbitrage trade orders andwaiting orders for stock 1.

Market Price Differential rate
market price differential ratio

0.12%
0.10%

0.08%
0.06%
0.04%
0.02%

100%

70%

50%

30%

20%

10%

0.00%

ETF order ratio (k)As ETF order ratio increased, market price differential ratio increased.
Even though more arbitrage trades occurred because of the larger ETF
order ratio, the market price differential rate did not improve.

What is a role of Simulation Models?
This book answer the question.
And also answer “What is amodel?”

Simulation and Similarity Using Models to Understand the World, 2012https://global.oup.com/academic /product/9780199933662

Aim is not replicating nor forecasting real world
In time, those Unconscionable Maps no longer satisfied,
and the Cartographers Guilds struck a Map of the Empire
whose size was that of the Empire, and which coincided
point for point with it . . . In the Deserts of the West, still
today, there are Tattered Ruins of that Map, inhabited by
Animals and Beggars;
On Exactitude in Science Jorge Luis Borges* Modeling, (is) the indirect study of real-world systems via theconstruction and analysis of models.
* Modeling is not always aimed at purely veridical representation.
Rather, they worked hard to identify the features of these systems
that were most salient to their investigations.
* Textbook model of the cell is both abstract and idealized relative
to any real cell. It is abstract because it isn’t a model of any
particular kind of cell; it is a model of properties shared by all
eukaryotic cells. Relatedly, it is idealized because its generality
forces some parts of the model to be distorted relative to any realcell. I think these are both interesting properties,

Which ‘’map’’ explains the access better?
Some of maps are models of the real geography for understanding an access.

Very different from the real,
however, very good explaining theaccess.

Very similar to the real, however,very bad explaining the access.

must shave non-investigating features from the model
Different investigations, different shaving parts.

Role of Model (in the case of Agent-Based Artificial Market Model)

InvestorA

Investor
B

Investor
C

Inherit Only Important Properties(Behaviors, Algorithms) for Investigating Phenomena

Other Investigating
Phenomena,
Other Important Properties,
Other Good Models

Never Real-Existing Investor
For Understanding Properties
of Real-Existing Investors

Model ofInvestors e.g.: Fashion Model: Understanding Closes
Model Home: Understanding Home

An Aim is to understand how Important Properties (Behaviors, Algorithms)
affect Investigating Macro Phenomena and play Roles in System.

It is NOT aim Replicating real-existing Investors A, B and C.
It is aim Understanding real-existing Investors.

Other Focusing Phenomena, Other Good Models

What simulation(computational) model can do and mathematical model can not do* When one invokes a computational model to explain some phenomenon,
one is typically using transition rules or algorithm as the explanans. Schelling
explained segregation by pointing out that small decisions reflecting small
amounts of bias will aggregate to massively segregated demographics. Neither
the time sequence of the model’s states nor the final, equilibrium state of themodel carries the explanatory force; the algorithm itself is needed.
Algorithms: The As want at least 30% of their neighbors to be As and likewise
for the Bs. An agent standing on some grid element e can have anywherefrom zero to eight neighbors in the adjoining elements.

Example of equation model cannot but simulation model can (Schelling Model)
Party with Students(#) and Professors(@)
#
# #
#
@
@ @
#
#
@

# @ #
# @
@
@
#
# @ # @
@ # @ @# # #@ # @ #
@

@
# @
@ #
# @
@
@
@
#

Micromotives and Macrobehavior, 2006
http://books.wwnorton.com/books/978-0-393-32946-9/

Rules:* around me (8pixs)
with more 1/3 myequals -> not move
* No -> move
After many steps,,,

#
#
#
@

#
#
#
@
@
@

@ @

#
# @
@ @
@
@ #
# #
# #

@ # #
@ @ #
@
@ @
@ @ @
# @ @
# #
#

#
#
@
@

Mod. rules:# and @ have been separated＃:need one more equal
The segregation occurs even if we want not to be@:need one less equal# # # #
# # # #
# # # #@ # @@ @ @ #
#
@ #
@
@

@
@ @

@
@
@

@
@
#
#
#

The space of #is smaller

@
@
@ @
@
# @
# #

heavily minority. We do NOT hate other kinds.

The purpose of simulation is understanding the reasonsand mechanism, not forecasting the final distribution.
Where are tables? How much they eat? Where areassistant professors? Is the party place square? Are
behaviors of people too simple?
---- Silly questions for the purpose of simulation.
prevent our understanding the mechanism.

Simplifying depends on that we want to understandSchelling recommended you simulate this manually using coins tounderstand what is simulation model.

We want know a relationship between
Micro Process:
Deciding Orders, Rules of Exchange& Macro Phenomena: Price Formation

Mathematical Model
Macro Model

Stock
MarketCountry Ａ
Bond
MarketCountry A
Stock
MarketCountry B

Replicate Only
Important Properties

can treat only this region

Model ofInvestors

Inherit Only
Important Properties

Role of Model (cont.)

Deciding
OrdersInvestorA
Investor
B

InvestorC

Model of (Algorithms)
Exchange
ChangePrice
Rules
Formation Matching(Simulation Orders
We want know
(Aggregation of
Result)a relationship
Algorithms)
between them