Dynamical Models
for Herd Behavior
in Financial Markets
I. Introduction
II. Model
- Markets
- Agents
- Links
III. Numerical results
IV. Conclusions
Sungmin Lee, Yup Kim
Kyung Hee University, Seoul, Korea
1
I. Introduction
Price return : R(ti )  ln P(ti )  ln P(ti 1 )
R /
S&P 500 high frequency data
of price return
2
V. Eguiluz, M. Zimmermann
PRL v85,5659 (2000)
Financial market
agent
link
Information cluster
1 a
: link
a
: Trade & isolated
Financial
crashes
a  0.30
a  0.10
a  0.01
3
II. Model
○ Markets
- Static : number of agents is fixed.
(Canonical Ensemble)
- Dynamic : number of agents is not fixed.
(Grand Canonical Ensemble)
Static market
M. Zimmermann
Dynamic market
Join
q
Trade
&
leave
4
○ Sort
of Agents
1. Normal agent
M. Zimmermann
2. Smart agent
5
○ Links
- Random attachment (Random Graph)
- Preferential attachment (Scale Free network)
Random graph
1
6
1
1


1
6
M. Zimmermann
Scale free


2 
18
36
1
2

2 
2
36
1
18
6
○ State of agent
- inactive state [waiting l  0 ]
- two active states [buying l  1 or selling l  1 ]
○ model rule
- with probability q , one agent participates in the
market
1 q
- with
, the network of links evolves dynamically
in the following way
j
(1) an agent
is selected at random
j
1 a
(2) with probability
, the state of j remains
inactive
and,instead, a new link between agent and other
agent a
j
(normal / smart agent , random graph / scale free
link)
(3) with , the state of
becomes active by
randomly
s(ti )
choosing the state 1 or -1 , and instantly all agents
belonging to the same cluster follow this same
action
○ Measurement
by imitation.
is measured. And all agents in
Aggregate
state
of
the system :
the
active cluster leave
si  the
s(ti )market.

l

l 1, N
- Price index : P(t )  P(t ) exp( s /  )
i 1
i
i
- Price return : R(ti )  ln P(ti )  ln P(ti 1 )
7
III. Numerical results
○ Static market
- Phase diagram
0.40
Normal / Random graph
Normal / Scale free
Smart / Random graph
Smart / Scale free
0.35
0.30
a
0.25
Non-crash
0.20
0.15
0.10
0.05
Crash
0.00
0
2000
4000
6000
8000
10000
N
** Zimmermann :
N  10 4
(Static market / Normal agent / Random graph)
8
○ Dynamic market
- The distribution of return R
Normal agent
1
RG (a=0.3)
RG (a=0.0005)
SF (a=0.3)
SF (a=0.0005)
crash(manipulation)
Prob(R)
0.1
q=0.1
0.01
1E-3
1E-4
1E-5
1E-5
1E-4
1E-3
R
0.01
0.1
1
Financial crash
Prob(R)
0.1
RG (a=0.3)
RG (a=0.01)
SF (a=0.3)
SF (a=0.01)
q=0.5
0.01
1E-3
C.E
1E-4
G.C.E
1E-5
1E-5
1E-4
1E-3
R
0.01
0.1
9
Smart agent
1
RG (a=0.3)
RG (a=0.0005)
SF (a=0.3)
SF (a=0.0005)
No crash
Prob(R)
0.1
q=0.3
0.01
1E-3
1E-4
1E-5
1E-5
1E-4
1E-3
R
0.01
0.1
1
Financial crash
Prob(R)
0.1
RG (a=0.3)
RG (a=0.01)
SF (a=0.3)
SF (a=0.01)
q=0.5
0.01
1E-3
1E-4
1E-5
1E-5
1E-4
1E-3
R
0.01
0.1
10
- The distribution of the normalized return
Normal agent
100000
RG (a=0.3)
RG (a=0.0005)
SF (a=0.3)
SF (a=0.0005)
10000
10000
Frequency
1000
1000
q=0.1
100
-1.0
-0.5
0.0
0.5
1.0
100
10
1
-10
-8
-6
-4
-2
0
R/
2
4
100000
8
10
RG (a=0.3)
RG (a=0.01)
SF (a=0.3)
SF (a=0.01)
10000
Frequency
6
q=0.5
1000
100
10
1
-20
-10
0
10
20
R/
11
Smart agent
RG (a=0.3)
RG (a=0.0005)
SF (a=0.3)
SF (a=0.0005)
10000
q=0.3
Frequency
1000
100
10
1
-10
-8
-6
-4
-2
0
R/
2
4
100000
8
10
RG (a=0.3)
RG (a=0.01)
SF (a=0.3)
SF (a=0.01)
10000
Frequency
6
q=0.5
1000
100
10
1
-20
-10
0
10
20
R/
12
- Phase diagram
X : no crash
△: subtle
a
q
0.1
0.3
0.5
a
q
0.1
0.3
0.5
⊙ : crash(manipulation)
○ : financial crash
0.3
0.1
0.01
0.005
0.001
x
x △ △ ⊙ ⊙
0.0005
x △ ⊙ ⊙ ⊙ ⊙
Exponential
decay
x △ ○
Power law
decay
0.3
0.1
0.01
x
x
x
x
<Normal agent>
0.005
0.001
0.0005
X △ △
x △ △ △ X
x △ ○
<Smart agent>
Exponential
decay
Power law
decay
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IV. Conclusions
◆ In
static market, a herd behavior is
enhanced by smart agents and
scale free type links.
◆ We suggest a model for a dynamic
herd behavior in financial market.
◆ In dynamic market, smart agents
didn’t show financial crash
at q  qc . (0.3  qc  0.5)
◆ The distribution of return shows
exponential decay at q  qc .
◆ The distribution of return shows
power law decay at q  qc .
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