Price Variation Limits
and Financial Market Bubbles:
Artificial Market Simulations
with Agents' Learning Process
Takanobu MIZUTA*
Kiyoshi IZUMISPARX Asset Management Co. Ltd.
The University of Tokyo
The University of Tokyo
CREST PRESTO, JST

Shinobu YOSHIMURA University of Tokyo* mizutata@gmail.com* http://www.geocities.jp/mizuta_ta/
http://www.slideshare.net/mizutata/cifer2013

1

Thank you very much. I’m Takanobu MIZUTA from SPARX Asset Management.
I’m also belonging to The University of Tokyo.
Today, I’m going to give a presentation under the title of This.

1

Introduction
market more
restrict trades Price Variation Limit
efficient or not？
？
out of certain
to Prevent
price ranges

OvershootExperiment 1
Replicate Overshoot?

Experiment 2
Parameters’
Condition
Artificial Market Model⊿Ppl, tpl
(Agent Based Simulation)

Verified byHazard Rate
NEED
Learning Process

NEED∆Ppl / t pl < v
Should Avoid∆Ppl / t pl << v
http://www.slideshare.net/mizutata/cifer2013

2■ Financial exchanges sometimes employ a “price variation limit”, which restrict
trades out of certain price ranges to avoid sudden large price fluctuations,“overshoots” overshoots occur in bubble and crash in real financial market■ There is a debate over whether the price variation limit makes financial market
more efficient or not.
■ To investigate price variation limits, we built an artificial market model
implementing a learning process.
■ Experiment 1, we will show that the model should be implemented the learning
process to replicate overshoot.
■ Experiment 2, We will also show that a parameters' condition of the pricevariation limit to prevent overshoot.

2

market more
restrict trades Price Variation Limit
efficient or not？
？
out of certain
to Prevent
price ranges

OvershootExperiment 1
Replicate Overshoot?

Experiment 2
Parameters’
Condition
Artificial Market Model⊿Ppl, tpl
(Agent Based Simulation)

Verified byHazard Rate
NEED
Learning Process

NEED∆Ppl / t pl < v
Should Avoid∆Ppl / t pl << v
http://www.slideshare.net/mizutata/cifer2013

3

First, I will describe our artificial market model.

3Artificial Market Model (Agent Based Model)
Chiarella et. al. [2009]● Continuous Double Auction⇒ to implement realistic price variation limit● Agent model is Simple⇒ to avoid arbitrary result “Keep it short and simple”
heterogeneous 1000 agents
Expected ReturnP
1  w1, j log ft + w2, j rht, j + w3, j ε tj ret, j =P∑i wi , j +wi , jStrategy
WeightFundamental Technical noise↑ Different
for each agent
Learning Process Our Original↑ need to replicate an overshootGood Performance Strategy wi , j is up
Bad Performance Strategy wi , j is down

4

We built an artificial market model on basis of Chiarella et. al. [2009].
●Pricing mechanism is Continuous Double Auction
It is not simple, but, we need to implement realistic price variation limit●Agent Model is Simple. This is to avoid arbitrary result, “Keep it short andsimple” principle.
We think Artificial Market Models should explain Stylized Facts as Simply as
possible.
There are heterogeneous 1000 agents. All agents calculate Expected Returnusing this equation,
And, the strategy weights are different for each agent
・First term is a Fundamental Strategy: When the market price is smaller than thefundamental price, an agent expects a positive return , and vice verse.
・Second term is a technical strategy: When historical return is positive, an agent
expects a positive return, and vice verse.
・Third term is noise.
Chiarella’s model did not include Learning Process, however,We built Learning Process of agents, this is our Original.
We showed that learning process need to replicate an overshoot.
Agents are comparing Historical Return and Each Strategy’s Return.
・When the strategy’s return and Historical Return are Same Sign, Good
Performance Strategy, The strategy’s Weight is Up.
・When the strategy’s return and Historical Return are Opposite Sign, Bad
Performance Strategy, The strategy’s Weight is Down.

4

market more
restrict trades Price Variation Limit
efficient or not？
？
out of certain
to Prevent
price ranges

OvershootExperiment 1
Replicate Overshoot?

Experiment 2
Parameters’
Condition
Artificial Market Model⊿Ppl, tpl
(Agent Based Simulation)

Verified byHazard Rate
NEED
Learning Process

NEED∆Ppl / t pl < v
Should Avoid∆Ppl / t pl << v
http://www.slideshare.net/mizutata/cifer2013

5

Next, I show simulation results of Experiment 1 about leaning process andreplicating overshoot5

Experiment 1： Learning and OvershootCase 1
Fundamental Value= constantCase 2Fundamental Value→ rapid increment(bubble trigger)

With
Learning Process

×
For
Each
Cases

Without
Learning Process6

We examined two Cases, Case 1, Fundamental Value is constant,
Case 2, Fundamental value is rapid incremented like this. This is bubble inducing
trigger.
For Each cases, we examined With learning process And WithOut learning
process.
Therefore, we examined four cases in all.

6

Case 1： Fundamental Value = constantcase 1
10100
10050
price

10000
9950
9900
non-learning

9850

learning

900000800000700000600000500000400000300000200000100000

0

9800

time

Small fluctuating around Fundamental Value

7

This Figure shows time evolution of market prices in case 1, Fundamental Valueis constant.
In both cases, With learning process and without learning process. the results arevery similar,
The prices were small fluctuating around Fundamental Value, here, Ten
Thousand7

Case 2： Fundamental Value → rapid increment (bubble trigger)
case 2
20000
18000
price

16000
14000
non-learninglearning
fundamental value

12000
10000

900000800000700000600000500000400000300000200000100000

0

8000

time

Only with learning process, Overshoot occurred
8This Figure shows time evolution of prices in case 2.
Fundamental value was changed at this time, increased to New Fundamental
Value, Fifteen Thousand.
This is the bubble inducing trigger.
Without Learning Process, Black line, Overshooting was not occurred.
On the other hand, with Learning Process, Red line,
the price went far beyond the new fundamental value.
Only with learning process, Overshoot occurred.

8

Traditional Stylized Factscase 1
non-learninglearning
3.018
5.394

kurtosisautocorrelation
coefficient forsquare return

lag
1
2
3
4
5
6
7
8
9

0.134
0.101
0.076
0.060
0.052
0.040
0.036
0.030
0.026

0.125
0.105
0.087
0.074
0.061
0.054
0.048
0.045
0.039

case 2
non-learninglearning
2.079
3.180
0.219
0.164
0.133
0.118
0.108
0.100
0.092
0.087
0.082

All cases replicated: Fat Tail and Volatility Clustering

0.325
0.293
0.274
0.261
0.253
0.247
0.241
0.237
0.238

9

This Table lists Traditional stylized facts in each case.
In all cases, both kurtosis and autocorrelation for square returns for all i arepositive.
This means that all cases replicate Traditional stylized facts: fat-tail and volatilityclustering.

9

Hazard Rate (similar to “run test”)
New Stylized fact to verify model replicating overshootH(i) conditional probability thatsequence of positive return ends at i, given that it lasts until i.
For Example i=3、H(3)Time・・・・ １

２

３

Return４
H(3)： Probability of
4th Return Negative

Empirical Studies:
Any cases: H(i) < 50%,
Overshoot period: H(i) decline with I rapidly
McQueen and Thorley [1994], Chan et. al. [1998]⇒ Overshoot returns tend to continue to be positive
this tendency stronger continuing positive returns longer
10

We propose Hazard Rate as New Stylized fact to verify model replicating
overshootHazard Rate Hi is conditional probability that sequence of positive return ends at i,given that it lasts until i.
For Example i=3, H3 means like this.
1st, positive return, 2nd, positive, 3rd positive,
In this condition, H3 is probability of 4th return become negative.
Empirical Studies showed that, Any cases, Hi for most of i are smaller than 50%
And when including overshoot period, Hi decline rapidly with i,This show that the overshoot returns tend to continue to be positive
And this tendency stronger continuing positive returns longer

10New Stylized Facts: Hazard Rate H(i)case 1
non-learninglearningcase 2
non-learninglearningihazard rate

1
2
3
4
5
6
7
8
9

56%
55%
55%
54%
54%
53%
52%
52%
53%

55%
52%
50%
49%
45%
44%
41%
40%
40%

56%
55%
53%
52%
48%
45%
40%
35%
30%

55%
50%
45%
40%
36%
29%
26%
22%
19%

Only with Learning process → Verified by Hazard Rate
And Only Case 2 with Learning ⇒ Replicating Overshoot11

This Table lists New Stylized Facts: Hazard Rate in each case.
In case 2 with learning, hazard rate declined rapidly.
This case can replicate a significant Overshoot like actual markets.
On the other hand, the case without learning, hazard rate dose not declined
rapidly.
The case can not replicate Overshoot.
Case 1, without learning, Hazard rates are upper 50% for all i.
This is Not consistent with empirical study.
On the other hand, Case 1, with learning, Hazard rates for most of i are smaller
than 50%, even when price fluctuations are stable.
This consistent with empirical study.
Therefore, only cases with Learning Process were verified by Hazard Rate,and only Case 2 can replicate overshoot.

11

Result Summary Experiment 1： Learning and Overshoot
Not-Consistent with
Consistent with
Empirical study
Empirical study↑↑
Without
With
Learning Process Learning Process
Case1
Fundamental Value= constantCase2
Fundamental Value→ rapid increment

Stable
Not-Verified byHazard Rate

StableVerified byHazard Rate

No-Overshoot
Not-Verified byHazard Rate

Overshoot(Bubble & Crush)
Verified byHazard Rate
12

Result Summary Experiment 1 relationship between Learning process andreplicating OvershootThe cases With learning process, both case 1 and case 2, were Consistent with
Empirical study verified by Hazard Rate.
And case 2 can replicate overshoot, bubble and crush
The cases Without Learning Process were Not consistent with Empirical study
Not verified by Hazard Rate.

12

market more
restrict trades Price Variation Limit
efficient or not？
？
out of certain
to Prevent
price ranges

OvershootExperiment 1
Replicate Overshoot?

Experiment 2
Parameters’
Condition
Artificial Market Model⊿Ppl, tpl
(Agent Based Simulation)

Verified byHazard Rate
NEED
Learning Process

NEED∆Ppl / t pl < v
Should Avoid∆Ppl / t pl << v
http://www.slideshare.net/mizutata/cifer2013

13

Next, I show Simulation Results about Price Variation Limit.

13

Experiment ２： Price Variation Limit
Price Variation Limit
Can Not order Outside P
Two ConstantParameters:t −t p l± ∆Pplt pl Limit time Span

∆Ppl Limit price Range
Buy order above herechanged toPt − t pl+ ∆Ppl∆Ppl

Pt − t pl

t pl

∆Ppl

Market PricePt − t plPt
Sell order under herechanged toPt − t pl− ∆Ppl14

We modeled the price variation limit like this.
There are two constant parameters.
Tpl is a limit time span, and ⊿Ppl is limit price range.
We referred market price Before tpl, Pt-tpl.
and any agents can Not order OutSide from Pt-tpl - ⊿Ppl to Pt-tpl + ⊿Ppl
Concretely,Any buy order prices above here, they are changed to this price.
and any sell order prices under here they are changed to this price.

14

Case 2 & with Learning, Price Variation Limitcase 2, learning
20000
18000

price

16000
14000
non Price Limit
Price Limit
fundamental value

12000
10000

900000800000700000600000500000400000300000200000100000

0

8000

time

Overshooting was vanished in price variation limit
But, price took longer to reach new fundamental price

15

This Figure shows time evolution of prices in case 2 with learning,
comparing the case implemented price variation limit and not implemented.
Overshoot was vanished In the case implemented price variation limit
However, price took longer to reach new fundamental price.
In the case not implemented price variation limit, price took faster to reach new
fundamental price.
Implemented case, slower to reach new fundamental value, converging is slower.

15

Overshooting Amplitude and Converging Speedcase 2, learning
20000

reach time to new
18000
fundamental value

max price fromnew fundamental value

price

16000
14000

non Price Limit
Price Limit
fundamental value

12000
10000

900000800000700000600000500000400000300000200000100000

0

8000

time

Preventing Overshooting and Immediate Converging→ no market achieves both at once
Optimization of t pl and ∆Ppl

16

Next, I investigated relationship between Overshooting Amplitude andConverging Speed.
I measured, here, reach time to new fundamental value, and here, max pricefrom new fundamental value.
We want Preventing Overshooting and Immediate Converging.
However, No market achieves both at once.
Therefore, it is important to Optimize of these Two parameters.

16

Max Price from New Fundamental Value
max price from new
fundamental value 1000
100
1,374
200
3,236
300
3,450
400
3,482
700
3,347
3,494
⊿Ppl 1000
1500
3,512
2000
3,454
2500
3,475
3000
3,436
4000
3,359
5000
3,556

Blue area:

tpl
2000
475
1,421
3,020
3,501
3,470
3,357
3,356
3,595
3,357
3,443
3,398
3,467

∆Ppl
t pl

3000
285
728
1,378
2,381
3,366
3,229
3,424
3,334
3,497
3,598
3,374
3,317

4000
205
485
900
1,307
3,311
3,578
3,404
3,317
3,368
3,431
3,607
3,509

7000 10000 15000 20000 25000 30000 40000 50000
115
76 ------------224
146
97
72 --------387
233
134
106
90
86 ----565
339
164
112
95
93
85 --1,395
734
363
211
120
94
90
83
2,633 1,388
681
421
267
122
93
91
3,407 3,019 1,384
607
286
208
184
183
3,682 3,493 2,408 1,094
580
452
411
418
3,262 3,466 3,524 2,007 1,442
508
103
100
3,268 3,330 3,365 2,921 1,602 1,190
980
931
3,673 3,378 3,338 3,351 3,433 1,701
477
401
3,440 3,162 3,486 3,311 3,320 3,231 1,345
104< v ≅ 0.128

v: Converging Speed
without price variation limit

Preventing Overshooting∆Ppl / t pl smaller (to upper right) ⇒ Overshoot smaller
NEED to prevent overshoots∆Ppl / t pl < v

17This table lists Max Price from New Fundamental Value in various Tpl and ⊿Ppl
Blue area, ⊿Ppl over Tpl is smaller than V
V is a converging speed without price variation limit, approximately 0.128.
As you see, in this area, max price small, this means preventing overshoot.
This is smaller, to upper right area, overshoot are smaller.
Therefore, we found that this is smaller than V is needed to prevent overshoot.

17

Reach time to New Fundamental Valuereach time to new
tpl
fundamental value
1000 2000 3000 4000 7000 10000 15000 20000 25000 30000 40000 50000(x 1000)100
55 104 157 216 382
555 ------------20040
55
79 103 181
261
385
509 --------30039
41
55
70 120
171
257
342
425
506 ----400
39
39
44
55
91
128
195
259
324
385
505 --700
40
39
40
39
55
75
113
148
185
229
302
371
39
39
39
39
41
54
78
104
127
150
193
234
⊿Ppl 1000
1500
39
39
39
39
39
40
54
72
93
113
146
175
2000
39
39
40
40
39
39
43
56
71
85
106
126
2500
39
39
39
39
39
39
39
46
53
60
74
86
3000
39
39
39
39
40
39
39
40
48
56
70
79
4000
39
39
39
39
39
39
40
39
39
44
63
76
5000
39
39
39
39
39
40
39
39
40
40
40
47

Blue area:

∆Ppl
t pl

< v ≅ 0.128

v: Converging Speed
without price variation limit∆Ppl / t pl smaller (to upper right) ⇒ converging speed slower
Should Avoid Not to be converging slower∆Ppl / t pl << v

18This table lists Reach time to New Fundamental Value in various Tpl and ⊿Ppl
Blue area, this is smaller than V.
This is smaller, to upper right area, converging speed are slower.
Therefore, we found that it this is too smaller than V is should be Avoid Not to beconverging slower.

18

Result Summary Experiment 2: about Price Variation Limit
Price Variation Limit
prevent Overshooting and Converging Slower
Optimization oft pl and ∆Ppl

∆Ppl / t pl smaller ⇒ Overshooting smaller
NEED to prevent overshoots∆Ppl / t pl < v∆Ppl / t pl << v

Should Avoid Not to be converging slower∆Ppl / t pl smaller ⇒ converging speed slower
v: Converging Speed
without price variation limit 19

Result Summary Experiment 2 about Price Variation Limit.
Price Variation Limit prevents Overshoot and cause Converging speed Slower to
new fundamental value
Therefore, it is important to Optimize of these two parameters.
When this is smaller, overshooting smaller.
We found that it needs smaller than, V to prevent overshoots
On the other hand, this is smaller, converging speed is slower.
We found that it should be avoid too smaller than V Not to be converging slower

19

Summary
market more
restrict trades Price Variation Limit
efficient or not？
？
out of certain
to Prevent
price ranges

OvershootExperiment 1
Replicate Overshot?

Experiment 2
Parameters’
Condition
Artificial Market Model⊿Ppl, tpl
(Agent Based Simulation)

Verified byHazard Rate
NEED
Learning Process

NEED∆Ppl / t pl < v
Should Avoid∆Ppl / t pl << v
http://www.slideshare.net/mizutata/cifer2013

20

I summarize this presentation■ To investigate price variation limits, we built an artificial market model
implementing a learning process.
■ Experiment 1, we showed that the model should be implemented the learning
process to replicate overshoot.
■ Experiment 2, We also showed that a parameters' condition of the pricevariation limit to prevent overshoot.

20

That’s all for my presentation.
Thank you very muchfor your cooperation !

http://www.slideshare.net/mizutata/cifer2013

21

Could you say that again? (もう一度、おっしゃっていただけますか？)I don’t quite understand your question. (ご質問の趣旨が良く分からないのですが)
Could you please rephrase your question? (ご質問を分かりやすく言い換えていただ
けますか)So, you are asking me about.... (つまり、お尋ねの内容は...ですね)
I totally agree with you. (私も全くあなたと同意見です)
That’s a very challenging question for me to answer. (それは私にとって非常に答えがいのある質問です)
That’s a question I’m not sure I can answer right now. (そのご質問にすぐお答えできるかどうか分かりません)
It would require further research. (さらなる研究結果を待ちたい)
You are right on that point. (その点に関してはあなたが正しい)
Our method will not solve the problem. (我々の方法ではその問題は解決できない)

21

Appendix

22

22

Artificial Market Model (Agent Based Model)
On basis of Chiarella et. al. [2009]+ Learning Process of agentscomparing between the case with Learning Process and without

Feature of our model○ agent model is Simple→ to avoid arbitrary result “Keep it short and simple”
Models should explain stylized facts as simply as possible● pricing mechanism is Continuous Double Auction→ not simple to implement realistic price variation limit☆ Learning process→ agents switch strategy, fundamental or technicalAn overshoot occurred in the case with the learning process
but did not occur in the case without the process23

We built an artificial market model on basis of Chiarella et. al. [2009].
Chiarella’s model did not include Learning Process, however,We built Learning Process of agents.
And we are comparing between the case with Learning Process and without it.
Our Agent Model is Simple. This is to avoid arbitrary result, “Keep it short andsimple” principle.
We think Artificial Market Models should explain Stylized Facts as Simply as
possible,
Our pricing mechanism is Continuous Double Auction
It is not simple, but, we need to implement realistic price variation limit
Learning process
Here, Learning process means agents are switching strategy, fundamental
strategy or technical strategy.
We will show that, an Overshoot occurred in the case With the learning process,however, overshoot did not occur in the case WithOut the process

23Agent Model
j: agent number (1000 agents)
ordering in number ordert: tick time
Expected Return

ret, j =wi , j and τ jRandom ofUniform Distributionτjrht, j = log( P t / P

t −τ j)TechnicalP
1  w1, j log ft + w2, j rht, j + w3, j ε tj P∑i wi , j Parameters for agentswi , jHistorical Return

i=1,3: 0～1
i=2: 0～10
0～10000

Fundamental
Pf Fundamental Price
10000 = constant
P t Market Price at t
Expected Price

noise

ε tj
Random ofNormalDistribution
Average=0
σ=3%Pet, j = P t exp(ret, j )24

Next, I will describe agent model.
All agents calculate Expected Return using this equation.
First term is a Fundamental Strategy:
When the market price is smaller than the fundamental price, an agent expects a
positive return , and vice verse.
Second term is a technical strategy:
When historical return is positive, an agent expects a positive return, and viceverse.
Third term is noise,
After the expected return has been determined, an expected price is determined
like this.
And, agents order base on this Expected Price.

24

Learning Process
Expected Return

ret, j =P1  w1, j log ft + w2, j rht, j + w3, j ε tj P∑i wi , j Historical Return

Compare each Strategy

rl = log P / Ptt

Same Sign
good performer Strategy
wi , j ← wi , j + krlt [0, ( wi , max − wi , j )]Weight Up
With 1% probability：
Resetwi , j ← [0, wi ,max ]t −t l

Opposite Sign
bad performer Strategy
wi , j ← wi , j − krlt [0, wi , j ]Weight Down[a,b]: Random Uniform Distribution
from a to b
25

We also developed a model implementing a learning process of agents.
Agents are comparing Historical Return and each Strategy’ term, Fundamental
strategy term, and Technical strategy term.
When the strategy’s expected return and Historical Return are Same Sign,
This means good performer Strategy.
The strategy’s Weight is Up.
When the strategy’s expected return and Historical Return are Opposite Sign,
This means bad performer Strategy.
The strategy’s Weight is Down.
We also added random learning.
In this way, agents learn better parameters and switch to the investment strategy
that estimates correctly.

25Order Price and Buy or Sell
price

Pet, j + Pd
Pt

Order Price Randomt
Uniformo , j （about±10%）PSell （one unit）

Pot, j > Pet, j

Expected Pricete, jBuy (one unit) P t < P t

P

o, j

e, j

Pet, j − Pd
To Stabilize simulation for continuous double mechanism,
Order Prices must be covered widely in Order Book.

26

Next, agents determine order price and, buy or sell.
To Stabilize simulation runs for the continuous double mechanism,
Order Prices must be covered widely in Order Book.
We modeled an Order Price, Po, by Random variables of Uniformly distributed in
the interval from Expected Price, Pe, minus constant, Pd, to Pe plus Pd.
And then,
When Po lager than Pe, the agent orders to sell one unit.
When Po smaller than Pe, the agent orders to buy one unit.

26

Hazard Ratecase 2 lerning
non Price Limit Price Limitihazard rate

1
2
3
4
5
6
7
8
9

55%
50%
45%
40%
36%
29%
26%
22%
19%

55%
53%
49%
47%
44%
42%
40%
36%
31%

Hazard Rates increased→ Result by preventing bubble

27

Hazard Rate
Hazard Rates increased in implemented price variation limit
This Result shows preventing bubble

27

Strategies Weight

900000800000700000600000500000400000300000200000100000fundamental weight (left axis)
technical weight (right axis)

100%
98%
96%
94%
92%
90%
88%
86%
84%
82%
80%

technical weight

10%
9%
8%
7%
6%
5%
4%
3%
2%
1%
0%
0

fundamental weightcase 2, learning

time

During overshooting, switching fundamental to technical
Consistent with empirical studies[Frankel and Froot, 1990], [Yamamoto and Hirata, 2012]28

I also show Strategies weight, in the case 2 with learningwhich case include overshoot.
During overshooting, agents are switching fundamental strategy to technicalstrategy.
This is consistent with empirical studies, like these studies.

28

Strategies Weight
100%
95%
90%
85%
fundamental weight (left axis)
technical weight (right axis)

80%

technical weight

10%
9%
8%
7%
6%
5%
4%
3%
2%
1%
0%

90000080000070000060000050000040000030000020000010000075%
0fundamental weightcase 2, learning, Price Limit

time

Agents Not switching to the technical strategy→ prevented overshooting.

29

I also show Strategies weight, in the case 2 with learning, implemented pricevariation limit
Agents are not switching fundamental strategy to technical strategy.
This causes preventing overshooting, bubble and crash.

29