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Evolutionary dynamics in financial markets with many trader types
Brock, W.A.; Hommes, C.H.; Wagener, F.O.O.
Link to publication
Citation for published version (APA):
Brock, W. A., Hommes, C. H., & Wagener, F. O. O. (2001). Evolutionary dynamics in financial markets with
many trader types. (CeNDEF working paper; No. 01-01). Amsterdam: University of Amsterdam.
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Download date: 29 Jul 2017
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3ihLc i @j2 5 r ' fc |i wAw t t)44i|hU h| |i u?_@4i?|@* t|i@_) t|@|i @?_ |i
wAw t |ihiuLhi ?L?}i?ihU W? |i t)44i|hU U@ti E}hi @c Mi|ii? |i OLTuc TihL_
_LM*?} E# @?_ T|UuLh! E6 MuhU@|L? Uhit |i u?_@4i?|@* t|i@_) t|@|i t
?^i @?_ t|@M*i t |i T@h@4i|iht UhLtt |i 6 Uhic |L @__|L?@* ?L?u?_@4i?|@*
t|i@_) t|@|it @hi Uhi@|i_ |ih hL|it |L ?t|@M*|) LUUh i? UhLtt?} |i # Uhic
ihi |i u?_@4i?|@* t|i@_) t|@|i MiUL4it ?t|@M*i @?_ @ Et|@M*i 2U)U*i t Uhi@|i_c
Lh i? UhLtt?} |i OLTu Uhic ihi |i u?_@4i?|@* t|i@_) t|@|i MiUL4it ?t|@M*i
@?_ @ Et|@M*i ?@h@?| UhU*i | TihL_U Lh ^@tTihL_U _)?@4Ut t Uhi@|i_ Ai
T|UuLh! MuhU@|L? Uhi t ?L?}i?ihU @?_ L?*) LUUht ? |i t)44i|hU U@ti `i?
|i t)44i|h) t MhL!i? M) @ ?L?3ihL 4i@? M@t >f 9' fc @t **t|h@|i_ ? }hi M
uLh >f ' fc |i 6 Uhi _t@TTi@ht @?_ Mhi@!t T ?|L |L }i?ihU UL_4i?tL? L?i
MuhU@|L? Uhitc @ OLTu @?_ @ t@__*i?L_i E5 MuhU@|L? Uhi `i? UhLtt?} |i 5
Uhi uhL4 Mi*Lc |L @__|L?@* t|i@_) t|@|it @hi Uhi@|i_c L?i t|@M*i @?_ L?i ?t|@M*i
L|Ui |@|c @t **t|h@|i_ ? }hi Uc i? |i Tih|hM@|L? t t4@** @t uLh >f ' fc
|i 5 @?_ |i OLTu Uhit @hi U*Lti |L |i 6 @?_ |i OLTu Uhit ? |i t)44i|hU
U@ti W? |t ti?ti |i MuhU@|L? _@}h@4 _iTi?_t UL?|?Lt*) L? |i T@h@4i|ihtc @?_
| t tiu* |L UL?t_ih |i t)44i|hU wAw @t @? RLh}@?3?} Ui?|ih |L t|_) MuhU@|L?
Ti?L4i?@ ? |i }i?ihUc ?L?t)44i|hU wAw
Ai 4Lt| hi*i@?| U@ti tii4t |L Mi |i U@ti i? |i 4i@? > Lu |i ht| Lh_ih ULiUi?|
if ? |i uLhiU@t|?} h*i Ee t@|tit f > W? |@| U@tic @t |i t|hU|h@* T@h@4i
|ih # ?Uhi@titc |i Eu?_@4i?|@* t|i@_) t|@|i *Ltit t|@M*|) |hL} @ OLTu MuhU@|L?
2S
E@ t)44i|hU U@ti E>f ' f
EM }i?ihUc ?L?t)44i|hU U@ti E>f ' f
2
2
1
Hopf
Hopf
0
mu1
0
mu1
SN
Sym 1:1
PF
1
−1
PD
−2
−1
PD
−2
−3
−3
1:2
−4
0
0.2
0.4
0.6
0.8
1:2
1
1.2
−4
0
1.4
0.2
0.4
0.6
eta
0.8
1
1.2
1.4
eta
EU ?L?t)44i|hU @?_ t)44i|hU U@tit
The effect of the perturbation on the bifurcation diagram
2
SN
1
Hopf
mu1
0
−1
PD
−2
−3
−4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
eta
6}hi G uhU@|L? _@}h@4t ? |i E#c > T@h@4i|ih T*@?i uLh wAw Eeec ihi >
hiThiti?|t |i 4i@? Lu |i ht| Lh_ih t|LU@t|U |hi?_ @h@M*i if ? |i uLhiU@t|?} h*i
Ee 6Lh >f ' @j2 ' fc | >f |i 4i@? Lu |i UL?t|@?| iff ? |i uLhiU@t|?} h*i Eec
|i wAw t t)44i|hU @?_ |t ?L?}i?ihU( i? >f 9' f |i wAw t ?L?t)44i|hU @?_
}i?ihU Ai _@}h@4t tL OLTu EOc TihL_ _LM*?} E#c T|UuLh! E6 @?_ t@__*i
?L_i E5 MuhU@|L? Uhit ? E#c > T@h@4i|ih T*@?ic | L|ih T@h@4i|iht i_ @|
- ' fc 5 r ' fc >2 ' > ' fc jf ' j ' j2 ' @?_ j ' f i|ii? |i OLTu @?_ #
Uhit E@?_ |i 6 Uhi i? >f ' f |ihi t @ ?^ic t|@M*i t|i@_) t|@|i At t|i@_)
t|@|i MiUL4it ?t|@M*i i? UhLtt?} |i OLTu Lh |i # Uhi MLi |i 6 Uhi Lh
|i 5 Uhi |i t)t|i4 @t |hii t|i@_) t|@|it Ai 6 Uhi t ?L?}i?ihU @?_ L?*)
@htit ? |i t)44i|hU U@ti | 4i@? M@t >f ' f `i? |i t)44i|h) t MhL!i? M)
Tih|hM?} |i 4i@? M@t |L >f ' fc |i 6 Uhi Mhi@!t< ?|L }i?ihU OLTu @?_ 5
Uhit
2.
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Uhi@tit W44i_@|i*) @u|ih |i OLTu MuhU@|L?c uLh # D E}hit e@Uc TihL_U
Lh ^@tTihL_U _)?@4Ut L? @ t|@M*i ?@h@?| UhU*i LUUht u|ih @ ^@tTihL_U OLTu
MuhU@|L?c uLh # ' D @?_ # ' D2c E^@tTihL_U _)?@4Ut L? @ t|@M*i ?@h@?|
|Lht LUUht +iU@** uhL4 |i iLti+i**iA@!i?t Ti?L4i?L? |@| 2 U*Lti |L @
t)t|i4 | @ ^@tTihL_U OLTu MuhU@|L? t)t|i4t | t|h@?}i @||h@U|Lht i t| Ai
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| |i wAw UL?ih}i?Ui |iLhi4 |ihiuLhi t}}it|t |@| Lh iL*|L?@h) t)t|i4t |
4@?) |h@_ih |)Tit U@? i M| t|h@?}i @||h@U|Lht 6}hit eu} tL |i ?t|@M*i 4@?
uL*_ Lu @ TihL_U t@__*i TL?|( |i Uh*?} t@Ti Lu |i ?t|@M*i 4@?uL*_ t}}it| ?i@h
L4LU*?U MuhU@|L?t @?_ Mhi@!?} T Lu |i ?@h@?| |Lht ?|L @ t|h@?}i @||h@U|Lh
6}hit e4 tL |@| ?Uhi@t?} # uh|ih *i@_t |L @ MuhU@|L? hL|i |L U@Lt @?_
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}hi e Thiti?|t ?4ihU@* i_i?Ui Lu |i LUUhhi?Ui Lu @| hLU! @?_ OL44it Ebb.
U@**i_ @ h@|L?@* hL|i |L h@?_L4?ittc |@| tc @ MuhU@|L? hL|i |L t|h@?}i @||h@U|Lht
@t |i ?|i?t|) Lu ULUi |L t|U uLhiU@t|?} t|h@|i}it ?Uhi@tit Wu tU h@|L?@* hL|it
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ThLM@M*|)
2H
1
1
-1
1
-1
-1
E@ # '
1
-1
-1
EM # ' e
1
1
-1
EU # ' D
1
-1
-1
-1
E_ # ' D
1
1.03743
-1.03468
-1.03468
-1
Ei # ' D2
0.3
E} # ' De
1
-0.06
-0.2
Eu # ' De
1.03743
2b
0.4
1
1.07309
-0.312659
-1.07304
-1.07304
-0.401489
E # ' DS2D
1.07309
1.07309
-1.07304
-1.07304
E # ' DS2D
-0.0500446
1.5
-1.5
E # ' D.
1.07309
1.17
-1.17
-1.17
-0.112291
-1.5
E! # ' Db
1.5
E4 # ' S
1.17
1.17
E* # ' Db
1.17
-1.17
-1.17
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3
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Time series − LTL simulation
3
3
2
2
1
1
returns (%)
returns (%)
Time series − S&P 500 1980−2000 without crash
0
0
−1
−1
−2
−2
−3
−3
0
500
1000
1500
2000
2500
3000
time (days)
3500
4000
4500
5000
0
500
1000
Distributions
1500
2000
2500
3000
time (days)
3500
4000
4500
5000
4000
4500
5000
Price deviation series of the LTL system
100
90
0.3
80
0.2
70
price deviation
probability density
0.1
60
50
40
0
−0.1
30
−0.2
20
−0.3
10
−0.4
0
−3
−2
−1
0
returns (%)
1
2
3
500
1000
0.2
0.15
0.15
0.1
0.1
0.05
0
−0.05
−0.05
10
20
30
40
50
lag
60
70
2500
3000
time (days)
3500
0.05
0
−0.1
2000
Correlations of returns and absolute returns
0.2
correlation coefficient
correlation coefficient
Correlations of returns and squared returns
1500
80
90
100
−0.1
10
20
30
40
50
lag
60
70
80
90
100
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1
0.8
0.6
0.4
f(x)
0.2
0
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−0.4
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−1
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0
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0.5
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3
3
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LTL − returns
1
1
0.8
0.8
normalized prediction error
normalized prediction error
S&P500 − returns
0.6
0.4
0.2
0
0.6
0.4
0.2
0
200
400
600
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number of neighbors
1400
1600
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0
0
200
400
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number of neighbors
1400
1600
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