---
title: An interaction between a leveraged ETF and futures in a crash investigated by an agent-based model
tags: 
author: [水田孝信](https://www.docswell.com/user/mizutata)
site: [Docswell](https://www.docswell.com/)
thumbnail: https://bcdn.docswell.com/page/LELK24WQJR.jpg?width=480
description: An interaction between a leveraged ETF and futures in a crash investigated by an agent-based model Takanobu Mizuta SPARX Asset Management Co. Ltd.  Note that the opinions contained herein are solely those of the authors and do not necessarily reflect those of SPARX Asset Management Co., Ltd.  In this study, the artificial market model was built by adding a market in which a L-ETF itself is traded and adding a erroneous orders leading turmoil implemented by the model of [Mizuta 2016] to the prior model of  [Yagi 2020]. We investigated that sharply falling of the futures or the L-ETF affects the another market and effects of rebalancing trades of the L-ETF.  In the result, the erroneous orders to the L-ETF leads that market prices of the L-ETF fall, but the arbitrage-trades reduce the fall. Here, waiting buy orders to the futures, not to the L-ETF, are executed and reduced because a sell order of a normal agent executes a buy order of the AA and then the AA sells the futures. The buy orders of the AA to the L-ETF reduce the fall of the L-ETF. The reason why the AA can buy the L-ETF is that  the AA can consume the  waiting buy orders of the futures. In short, liquidity of the futures is consumed to prevent the fall of the L-ETF. This means that liquidity of the futures is used as hidden liquidity of the L-ETF.  In the case that the erroneous orders to the futures, the arbitrage-trades by the AA mix and combine liquidity of the L-ETF and futures, therefore, same falling happens even which to the L-ETF or the futures the erroneous orders come. The futures can gain only a less liquidity from the L-ETF that has a less liquidity, then this leads not to reduce falling so much. In real financial market, this means that investors of L-ETFs have great benefit by gaining much liquidity of futures, on the other hand, investors of futures do not have so benefit by gaining low liquidity of L-ETFs.  Larger the L-ETF, larger falling for both the L-ETF and the futures. When S0 &gt; 100000, the simulation can not run due to too large falling of market prices. This means that markets are destroyed. When the futures fall, the L-ETF sells futures. Here, when the L-ETF is so big, the futures also fall, and then the number of futures the L-ETF should sell is larger, finally, the L-ETF should additionally sell the futures. Such process continues as loop and falling can not stop. This leads to destroy the markets.
published: December 12, 24
canonical: https://www.docswell.com/s/mizutata/K6VER7-2024-12-12-121108
---
# Page. 1

![Page Image](https://bcdn.docswell.com/page/LELK24WQJR.jpg)

22-23 October 2024
2024 IEEE Computational Intelligence
for Financial Engineering and Economics (IEEE CIFEr)
https://www.stevens.edu/page-right-nav/cifer-conference-2024
https://www.stevens.edu/page-right-nav/cifer-conference-2024
An interaction between a leveraged ETF and futures in
a crash investigated by an agent-based model
Takanobu Mizuta SPARX Asset Management Co. Ltd.
mizutata[at]gmail.com https://mizutatakanobu.com
mizutata[at]gmail.com https://mizutatakanobu.com
Isao Yagi
Kogakuin University
Note that the opinions contained herein are solely those of the authors and do not
necessarily reflect those of SPARX Asset Management Co., Ltd.
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
1


# Page. 2

![Page Image](https://bcdn.docswell.com/page/4JM28KYKJW.jpg)

(1) Introduction
(2) Our Model
(3) Simulation Results
(4) Conclusion
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
2


# Page. 3

![Page Image](https://bcdn.docswell.com/page/PJRM5X96E9.jpg)

(1) Introduction
(2) Our Model
(3) Simulation Results
(4) Conclusion
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
3


# Page. 4

![Page Image](https://bcdn.docswell.com/page/PEXVK6QD7X.jpg)

Leveraged ETF (L-ETF)
ETF = Exchange Traded Funds is an investment fund
traded on stock exchanges, much like stocks
A Leveraged ETF is designed to deliver several times the return of its benchmark
index (future), e.g. S&amp;P 500, Euro Stoxx 50, FTSE 100, Nikkei 225
140
Index (Future)
Leveraged ETF (x2)
Value
120
100
80
60
0
2
4
6
8 10 12 14
days
Actually, Leveraged ETF has to trade the
Rebalance
index on daily to maintain its leverage
4


# Page. 5

![Page Image](https://bcdn.docswell.com/page/3EKQ519DED.jpg)

Rebalance to maintain its leverage
For example, when the index returns are +10% on day 1 and -10% on day 2
The Leverage is twice (x2)
Index
Days
0
1
2
(a) Return
+10%
-10%
Leveraged ETF (x2)
is Designed as
(b) Return
= 2x(a)
+20%
-20%
Leveraged ETF (x2) Holding
Index (future) Value
(c) Value
Will be by (b)
+20%
-20%
$100
$120
$96
(d) Should
have = 2x(c)
x2
x2
x2
(e) Will be
by (a)
$200 +10% $200
$240
$220
$192 -10% $216
So, leveraged ETF
Buy the futures when its price goes up
Sell the futures when its price goes down
This leads
Momentum
Trading
This momentum trading Rebalance concern some people,
This Rebalance makes Financial Market Unstable?
(f) Rebalance
= (d)-(e)
+20
-24


# Page. 6

![Page Image](https://bcdn.docswell.com/page/L73PKQWPE5.jpg)

Difficulty of Empirical Study
There are many empirical studies that produce both opposite results
Difficulty of Empirical Study
Empirical studies cannot be conducted to isolate the pure
contributions of L-ETFs due to the many diverse factors
affecting price formation in actual markets
Artificial Market Model
can
agent-based model for a financial market
6


# Page. 7

![Page Image](https://bcdn.docswell.com/page/87DL39K3EG.jpg)

An artificial market model = an agent-based model for a financial market
Virtual and Artificial financial Market built on Computers
Models
Include
Agents (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 and
determines Market Price
Determination market price
Complete Computer Simulation needing NO Empirical Data
✓ can isolate the pure contributions of L-ETFs
✓ can be conducted to investigate situations that have never
occurred in actual financial markets
✓ can effectively handling micro-macro feedback loops
7


# Page. 8

![Page Image](https://bcdn.docswell.com/page/VJP24ZKPJ8.jpg)

Previous Studies by Artificial Market Models
[Yagi 2016]
When a L-ETF impacts market prices more than their volatility
Market will be crushed
[Yagi 2020]
Using a continuous double auction
Investigated rebalance-trades to less impact to market prices
However, no previous study implemented a market in which a L-ETF
itself is traded in an artificial market model and investigated effects of
falling market prices of L-ETFs to those of futures.
Therefore in this study
[Yagi 2020]
+
L-ETF market
Arbitrage Agent
We investigated that sharply falling of the futures or the L-ETF affects
the another market and effects of rebalancing trades of the L-ETF
8


# Page. 9

![Page Image](https://bcdn.docswell.com/page/2EVDXWVVEQ.jpg)

(1) Introduction
(2) Our Model
(3) Simulation Results
(4) Conclusion
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
9


# Page. 10

![Page Image](https://bcdn.docswell.com/page/57GYVDL17L.jpg)

Overall view of our model
Normal agents
(NAs)
1000 agents
trade only
the futures
trades
place an order
with100%
High liquidity
An exchange
for
the futures
Arbitrage
Agent (AA)
Only one
NAs
1000 agents
trades only
The L-ETF
place an order
with probability 20%
Low liquidity
trades
An exchange
for
the L-ETF
rebalance
-trades
holding
the futures
L-ETF
agent
Only one
Holding the futures
10


# Page. 11

![Page Image](https://bcdn.docswell.com/page/4EQM61YNJP.jpg)

Continuous Double Auction
Waiting Orders
Shares
Sell
10
30
50
130
When sell order come here
transaction immediately occurs
Price
103
102
101
100
99
98
97
96
Shares
Buy
150
When buy order come here
transaction immediately occurs
70
Multiple buyers and sellers compete to buy and sell stocks in the market, and
transactions can occur at any time whenever an offer to buy and an offer to
sell match.
11


# Page. 12

![Page Image](https://bcdn.docswell.com/page/KJ4V4RW3E1.jpg)

Normal Agents (NAs)
This term is needed To
replicate stylized facts
1000 agents
j: agent number ordering in number order
t: tick time
Expected Return
of each agent
Technical
𝑡−1
𝑃
1
𝑃
𝑓
𝑡
𝑟𝑒,𝑗
=
𝑤1,𝑗 log 𝑡−1 + 𝑤2,𝑗 log 𝑡−𝜏𝑗 + 𝑤3,𝑗 𝜀𝑗𝑡
σ𝑖 𝑤𝑖,𝑗
𝑃
𝑃
noise
wi , j and  j
Fundamental
Pf Fundamental Price
 tj
Random of
Uniform Distribution
P t Market Price at t
Parameters for agents
wi , j
j
i=1,3: 0～1
i=2: 0～100
0～10000
Random of
Normal
Distribution
Average=0
σ=3%
10000 = constant
This term is needed to
prevent the prices go
out far away
To keep agents varied
To keep simulation runs stably
Expected Price 𝑡
𝑃 𝑒,𝑗 = 𝑃𝑡 exp( 𝑟 𝑡 𝑒,𝑗 )
of each agent
12


# Page. 13

![Page Image](https://bcdn.docswell.com/page/LE134QYZJG.jpg)

Fundamental Strategy
Fundamental Price ＞ Market Price -&gt; Expect + return
Fundamental Price &lt; Market Price -&gt; Expect - return
Technical Strategy (Historical Return)
Historical Return ＞ 0 -&gt; Expect + return
Historical Return &lt; 0 -&gt; Expect - return
Technical
Strategy
Fundamental
Price
Market
Price
Fundamental
Strategy
13


# Page. 14

![Page Image](https://bcdn.docswell.com/page/GEWPXMG6J2.jpg)

Erroneous orders
Erroneous orders period
30000-60000
Dropping
depth
time
Within the period, with a constant
probability 8% for the future, 40% for the L-ETF
&lt;- same total erroneous orders
each order of the NAs is changed to sell with very low price
Increasing such erroneous sell orders is what makes market prices fall
14


# Page. 15

![Page Image](https://bcdn.docswell.com/page/47ZK6QLR73.jpg)

Arbitrage Agent (AA)
Frequency of Trades
AA
t=1
t=2
t=3
t=4
t=5
time
NAs
The AA places an order and cancel an order any time
15


# Page. 16

![Page Image](https://bcdn.docswell.com/page/YJ6Q26W1JV.jpg)

Price Relationship between L-ETF and Futures
Process of an arbitrage in real financial markets is very complex, e. g.,
a ratio that investors can equivalent-exchange the L-ETF for the
futures are pass-depended of time evolution of their market prices.
In this study, the process of an arbitrage was modelized very simply
The Leverage is twice (x2)
L-ETF Return = 2 × Futures Return
Fundamental P
Futures
L-ETF
This means
10000
5000
Market Prices
Price Change
Range
10000
about 5000
---
about
same as futures
Example of Price
Change
+500
+500
same
L-ETF Price = Futures Price - 5000
Arbitrage Agent
Example of
Return
5%
10%
Twice
place orders to maintain the price relationship
16


# Page. 17

![Page Image](https://bcdn.docswell.com/page/GJ5R26ML74.jpg)

Immediately executed
arbitrage-trades
L-ETF
Sell Price
7
5030
10
5020
7
5010
12
5000
4990
Best Sell Price 8
Best Buy Price 4980
4970
4960
Buy
AA
10
6
4
Futures
Sell
Price Buy
30
10400
44
10300
70
10200
134 10100
10000 120
9900
88
9800
52
9700
25
Usually best sell price &gt; best buy price, rarely can be reversed when
orders from two different exchanges are combined
AA can earn to buy at the best sell price and sell at the best buy price
If
L-ETF best Sell + 5000 ＜ Future best Buy
Buy L-ETF Sell Futures
If
Futures best Sell - 5000 ＜ L-ETF best Buy
Buy Futures Sell L-ETF
Taking all arbitrage opportunities at that moment
17


# Page. 18

![Page Image](https://bcdn.docswell.com/page/9E2D4M93ER.jpg)

Arbitrage-trades
by waiting orders
Sell
7
10
L-ETF
Price
5030
5020
5010
5000
4990
4980
4970
4960
Buy
1
10
6
4
Wait AA
Sell
30
44
70
134
Immediately
Sell
After waiting
order executed
Futures
価格
Buy
10400
10300
10200
10100
10000 120
9900
88
9800
52
9700
25
AA places the waiting order to earn by immediately
executing for another asset
If
L-ETF best Buy &lt; Futures best Buy - 5000 ＜ L-ETF best Sell
Placing buy waiting order at Futures best Buy – 5000 on L-ETF
When the waiting order is executed, AA sells Futures
L-ETF best Buy &lt; Futures best Sell - 5000 ＜ L-ETF best Sell, vice versa
18


# Page. 19

![Page Image](https://bcdn.docswell.com/page/D7Y2MX487M.jpg)

L-ETF Agent
The L-ETFA decides every 10
if it rebalance-trades
t=10
Holding
Futures
L-ETF Agent
t=20
t=30
t=40
time
NAs
To maintain leverage twice, it should holds
S0×(Future Price/10000)
It trades when different more than 1%
Buy the futures when its price goes up
Sell the futures when its price goes down
This Rebalance makes
Financial Market Unstable?
19


# Page. 20

![Page Image](https://bcdn.docswell.com/page/VEN8W1Y9J8.jpg)

(1) Introduction
(2) Our Model
(3) Simulation Results
(4) Conclusion
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
20


# Page. 21

![Page Image](https://bcdn.docswell.com/page/Y79KX5P373.jpg)

5200
Time evolution of L-ETF prices
Liquidity of the futures is
consumed to prevent the fall
of the L-ETF
5000
Market prices
AA reduces the fall
5100
4900
4800
4700
4600
4500
4400
4300
140000
130000
120000
110000
90000
100000
Time
80000
70000
60000
50000
40000
30000
20000
10000
0
Rebalance-trades make enlarge
the fall of market prices
4200
Without either the erroneous orders nor the AA
Without the AA
Without the rebalance-trades
With the rebalance-trades(S0=50000)
21


# Page. 22

![Page Image](https://bcdn.docswell.com/page/G78R2YDLED.jpg)

Arbitrage
NAs
Sell
7
10
7
12
8
L-ETF
Price
5030
5020
5010
5000
4990
4980
4970
4960
Buy
AA
10
6
4
Futures
Sell
Price Buy
30
10400
44
10300
70
10200
134 10100
10000 120
9900
88
9800
52
9700
25
NAs sell L-ETF -&gt; AA buy L-ETF -&gt; AA Sell Futures
So, NAs sell futures indirectly
Liquidity of the futures is consumed to prevent the fall of the L-ETF
Liquidity of the futures is used as hidden liquidity of the L-ETF
22


# Page. 23

![Page Image](https://bcdn.docswell.com/page/L7LK24MQER.jpg)

Market prices of the futures
140000
130000
120000
9200
110000
4200
100000
9400
90000
4400
80000
9600
70000
4600
60000
9800
50000
4800
40000
10000
30000
5000
20000
10200
10000
5200
0
Market prices of the L-ETF
Time evolution of L-ETF and Futures prices with AA
Time
The L-ETF (left)
The futures (right)
The two lines overlap each other
when the market prices of the L-ETF change double
with those of the futures
23


# Page. 24

![Page Image](https://bcdn.docswell.com/page/4EM28K6KEW.jpg)

The amount of rebalance-sell orders
50000
20000
10000
5000
2000
1600
1400
1200
1000
800
600
400
200
0
0
Larger the L-ETF (larger S0),
larger falling for both
the L-ETF and the futures
800
700
600
500
400
300
200
100
0
without the AA
Consuming liquidity of the
futures reduces falling of
the L-ETF,
on the other hand, the
futures also fall.
Falling depth
Case with the erroneous orders to the L-ETF for several S0
S0(the initial number of futures the L-ETFA holds)
Falling depth of the L-ETF (left)
Destroying markets with
too big L-ETF (S0&gt;100000)
Falling depth of the futures (left)
The amount of rebalance-sell orders of the AA (right)
24


# Page. 25

![Page Image](https://bcdn.docswell.com/page/PERM5XP679.jpg)

1000
400
800
300
600
200
400
100
200
0
0
Investors of L-ETFs have
great benefit by gaining much
liquidity of futures.
On the other hand,
Investors of futures do not have
The amount of rebalance-sell orders
500
50000
1200
20000
600
10000
1400
5000
700
2000
1600
0
But with AA the results are
almost same as the
previous case
800
without the AA
Without AA,
The falling is not so deep
than the case of L-ETF.
Erroneous orders to
Futures are less than L-ETF
Falling depth
Case with the erroneous orders to the Futures for several S0
S0(the initial number of futures the L-ETFA holds)
Falling depth of the L-ETF (left)
Falling depth of the futures (left)
The amount of rebalance-sell orders of the AA (right)
25


# Page. 26

![Page Image](https://bcdn.docswell.com/page/P7XVK63DJX.jpg)

Mechanism of destroying markets with too big L-ETF (S0&gt;100000)
The L-ETF
rebalancesells futures
The number of
futures the L-ETF
should sell is larger
The
futures
fall
The
futures fall
more
The L-ETF
rebalancesells futures
When the L-ETF is so big,
the sells largely impact to
market prices of the futures
Sells largely impact to market prices, the futures fall more,
the L-ETF should additionally sell the futures
Such process continues as loop and falling can not stop
This result is consistent with [Yagi 2016]
26


# Page. 27

![Page Image](https://bcdn.docswell.com/page/37KQ51YD7D.jpg)

(1) Introduction
(2) Our Model
(3) Simulation Results
(4) Conclusion
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
27


# Page. 28

![Page Image](https://bcdn.docswell.com/page/LJ3PKQ9P75.jpg)

Summary (1/2)
✓ In this study, the artificial market model was built by adding a market in which a L-ETF
itself is traded and adding a erroneous orders leading turmoil implemented by the model
of [Mizuta 2016] to the prior model of [Yagi 2020]. We investigated that sharply falling
of the futures or the L-ETF affects the another market and effects of rebalancing trades
of the L-ETF.
✓ In the result, the erroneous orders to the L-ETF leads that market prices of the L-ETF fall,
but the arbitrage-trades reduce the fall. Here, waiting buy orders to the futures, not to
the L-ETF, are executed and reduced because a sell order of a normal agent executes a
buy order of the AA and then the AA sells the futures. The buy orders of the AA to the LETF reduce the fall of the L-ETF. The reason why the AA can buy the L-ETF is that the
AA can consume the waiting buy orders of the futures. In short, liquidity of the futures
is consumed to prevent the fall of the L-ETF. This means that liquidity of the futures is
used as hidden liquidity of the L-ETF.
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
28


# Page. 29

![Page Image](https://bcdn.docswell.com/page/8JDL39G37G.jpg)

Summary (2/2)
✓ In the case that the erroneous orders to the futures, the arbitrage-trades by the AA mix
and combine liquidity of the L-ETF and futures, therefore, same falling happens even
which to the L-ETF or the futures the erroneous orders come. The futures can gain only
a less liquidity from the L-ETF that has a less liquidity, then this leads not to reduce
falling so much. In real financial market, this means that investors of L-ETFs have great
benefit by gaining much liquidity of futures, on the other hand, investors of futures do
not have so benefit by gaining low liquidity of L-ETFs.
✓ Larger the L-ETF, larger falling for both the L-ETF and the futures. When S0 &gt; 100000,
the simulation can not run due to too large falling of market prices. This means that
markets are destroyed. When the futures fall, the L-ETF sells futures. Here, when the LETF is so big, the futures also fall, and then the number of futures the L-ETF should sell
is larger, finally, the L-ETF should additionally sell the futures. Such process continues as
loop and falling can not stop. This leads to destroy the markets.
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
29


# Page. 30

![Page Image](https://bcdn.docswell.com/page/VEP24Z3PE8.jpg)

Future Works
✓ The difference in results in the case of erroneous orders to the futures or L-ETF may
depend only on the lower liquidity. This is a future work.
✓ Note that an arbitrage in real financial markets is extremely more complex than our
model. In the actual, market prices of L-ETFs do not track preciously like as our model.
Some process existing in real financial market but not included in our model could be
important. All of them are future works.
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
30


# Page. 31

![Page Image](https://bcdn.docswell.com/page/27VDXW4V7Q.jpg)

References
[Yagi 2016] Yagi, I., Mizuta, T., “Analysis of the Impact of Leveraged ETF Rebalancing Trades on the Underlying Asset Market Using Artificial Market Simulation”, 12th Artificial Economics Conference, September 20-21, 2016
http://ae2016.it/public/ae2016/files/ssc2016_Mizuta.pdf
⚫ [Yagi 2016] Yagi, I., Mizuta, T., “Analysis of the Impact of Leveraged ETF Rebalancing Trades on the
Underlying Asset Market Using Artificial Market Simulation”, 12th Artificial Economics Conference,
September 20-21, 2016 http://ae2016.it/public/ae2016/files/ssc2016_Mizuta.pdf
[Yagi 2020] Yagi, I., Maruyama, S., Mizuta, T., “Trading strategies of a leveraged ETF in a continuous double auction market using an agent-based simulation”, Complexity, 2020
https://doi.org/10.1155/2020/3497689
⚫ [Yagi 2020] Yagi, I., Maruyama, S., Mizuta, T., “Trading strategies of a leveraged ETF in a continuous
double auction market using an agent-based simulation”, Complexity, 2020
https://doi.org/10.1155/2020/3497689
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
31


# Page. 32

![Page Image](https://bcdn.docswell.com/page/5JGYVD11JL.jpg)

References (Reviews of agent-based models for a financial market)
Review (2020)
Mizuta
https://arxiv.org/abs/1906.06000
of an agent-based
An agent-based
model
model
for designing
for designing
a financial
a financial
market
market that works well, CIFEr 2020 arXiv
Review of an agent-based model for designing a financial market
Mizuta (2020) An agent-based model for designing a financial market that works well, CIFEr 2020
arXiv https://arxiv.org/abs/1906.06000
Slide: https://mizutatakanobu.com/2021kyushu.pdf
YouTube: https://youtu.be/rmlb72ykmlE
Slide: https://mizutatakanobu.com/2021kyushu.pdf
YouTube: https://youtu.be/rmlb72ykmlE
Mizuta (2022) Artificial Intelligence (AI) for Financial Markets: A Good AI for Designing Better Financial Markets and a Bad AI for Manipulating Markets https://doi.org/10.1007/978-981-19-0937-5_13
Mizuta (2022) Artificial Intelligence (AI) for Financial Markets: A Good AI for Designing Better Financial
Markets and a Bad AI for Manipulating Markets https://doi.org/10.1007/978-981-19-0937-5_13
Citing many
Mizuta
https://ssrn.com/abstract=2710495
(2016)
previous
A Brief studies
Review of Recent Artificial Market Simulation Studies for Financial Market Regulations And/Or Rules, SSRN Working Paper Series
Citing many previous studies
Mizuta (2016) A Brief Review of Recent Artificial Market Simulation Studies for Financial Market
Regulations And/Or Rules, SSRN Working Paper Series
https://ssrn.com/abstract=2710495
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
32


# Page. 33

![Page Image](https://bcdn.docswell.com/page/47QM61DNEP.jpg)

Appendix
You can download this presentation material from
https://mizutatakanobu.com/2024CIFEr.pdf
https://mizutatakanobu.com/2024CIFEr.pdf
33


# Page. 34

![Page Image](https://bcdn.docswell.com/page/KE4V4RZ371.jpg)

Order Price and Buy or Sell
Price
Order Price of each NA
P t o, j
Sell(one unit)
Random
(Uniformly
Distribution)
Expected Price of each NA
P e , j = P exp( r e , j )
t
t
t
Buy(one unit)
To replicate many waiting limit orders,
order price is scattered around expected price
NA places one buy order when order price &gt; expected price
NA places one sell order when order price &lt; expected price
34


# Page. 35

![Page Image](https://bcdn.docswell.com/page/L7134QRZEG.jpg)

Verification: Stylized Facts
The purpose of simulation is understanding the reasons and mechanism,
not replicating ALL Stylized Facts
The simplicity of the model is very important because unnecessary replication of macro 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 in related factors.
Many empirical studies, e.g., Sewell 2006 have shown that both stylized facts (fat-tail and volatilityclustering) exist statistically in almost all financial markets.
Conversely, they also have shown that only the fat-tail and volatility-clustering are stable for any asset
and in any period because financial markets are generally unstable.
Fat-tail
1 to 100
kurtosis of price returns is positive
Volatility-clustering 0 to 0.2
square returns have a positive auto-correlation
The magnitudes of
these values are
unstable and vary
greatly depending on
the asset and/or
period.
For the above reasons, an artificial market model should replicate these values as significantly positive
and within a reasonable range as I mentioned. It is not essential for the model to replicate specific
values of stylized facts because the values of these facts are unstable in actual financial markets.
35


# Page. 36

![Page Image](https://bcdn.docswell.com/page/G7WPXM16E2.jpg)

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).
The simplicity of the model is very important for this study, because unnecessary replication of macro 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 in related factors.
36


