financial networks with static and dynamic thresholds tian qiu nanchang hangkong university
Post on 20-Dec-2015
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2
Outline
Motivation Financial networks with static and dy
namic thresholds Topology dynamics Economic sectors Conclusions
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Motivation
We introduce a dynamic financial network with both
static and dynamic thresholds based on the daily data
of the American and Chinese stock markets, and
investigate the topology dynamics, such as the
average clustering coefficient, the average degree and
the cross correlation of degrees. Special attention is
focused on dynamic effect of the thresholds on the
network structure and network stability.
5
Financial networks with static and dynamic thresholds
)(ln)(ln),( tPttPttR iii
We define the price return
22iii RR
i
iii
RRtr
)(
We normalize the price return to
where
6
Financial networks with static and dynamic thresholds
We define an instantaneous equal-time cross-correlation between two stocks by
)()()( trtrtG jiij
take individual stocks as nodes and set a threshold to create
links. At each time step, if the cross correlation ,
then add a link between stocks i and j ; otherwise, cut the link.
)(tGij
7
Financial networks with static and dynamic thresholds
static threshold
1
1 1 1
)()1(
2 N
i
N
ij
T
tijs tG
TNNQ
8
Financial networks with static and dynamic thresholds
1
1 1
)()1(
2)(
N
i
N
ijijd tG
NNtQ
dynamic threshold
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Topology dynamics detrended fluctuation analysis(DFA)
Average clustering coefficient
Average degree
cross correlation of degrees
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Topology dynamics-detrended fluctuation analysis(DFA)
For a time series A(t’’), we eliminate the average trend from the time series by introducing
'
1'']''([)'(
t
t aveAtAtB
Uniformly dividing [1, T ] into windows of size t and fitting B(t’) to a linear function in each window, we define the DFA function as
)'(tBt
T
tt tBtB
TtF
1'
2)]'()'([1()(
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Topology dynamics-detrended fluctuation analysis(DFA)
0.1
5.00 5.0
In general, F(t) will obey a power-law scaling behavior
indicate anti-correlated time series
ttF ~)(
0.15.0 0.1
indicate long-range correlating time series
indicate the Gaussian white noise
indicate noise
indicate unstable time series
f/1
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Topology dynamics-Average clustering coefficient
where is the clustering coefficient of node
N
ii tcN
tC1
)(1
)(
The average clustering coefficient is defined by
i)(tci
15
Topology dynamics-Average degree
where is the degree of node
N
ii tkN
tK1
)(1
)(
The average degree is defined by
i)(tki
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Why is the dynamic threshold crucial?
One important reason is that the volatilities fluctuate strongly in the
dynamic evolution, especially on the crash days. It induces large t
emporal fluctuations of the cross correlations of price returns. Thu
s the static threshold leads to dramatic changes in the topological
structure of the network. However, the dynamic
threshold proportional to suppresses such kinds of fluctuation
s and results in a stable topological structure of the network.
)(tQd
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Why is the dynamic threshold crucial?
74K
Extreme market
(30 days)
Stable market
(30 days)
Static threshold
Dynamic threshold
166eK 56sK
108K
109sK102eK
78.0C
88.0C
95.0eC 73.0sC
83.0eC 89.0sC
108K
78.0C74K
88.0C
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Topology dynamics-cross correlation of degrees
The so-called assortative or disassortative mixing on the degrees refers to the cross correlation of degrees. ‘Assortative mixing’ means that high-degree nodes tend to directly connect with high-degree nodes, while ‘disassortative mixing’ indicates that high-degree nodes prefer to directly connect with low-degree nodes.
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Topology dynamics-cross correlation of degrees
where and are the degrees of the nodes at both ends of the link, with
21221
211
)](21
[)(21
)](21
[)(
kjMkjM
kjMkjMtr
The cross correlation of degrees is defined as
j k
th M,...,1
,0r 0r,0r represent assortative mixing, no assortative mixing and disassortative mixing, respectively.
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Topology dynamics- Economic sectors
ikiii kkk /)(~
)(~tki
we first introduce the normalized individual degrees
We then construct the cross correlation matrix F of individual degrees whose elements are
)(~)(
~1
1
tktkT
FT
tjiij
and compute its eigenvalues and eigenvectors.
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Topology dynamics- Economic sectors
sQ
A:basic materials; B: conglomerates; C: consumer goods; D: finance; E: healthcare;F: industrial goods; G: services; H: technology; I: utilities.
)(tQd
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Conclusions the dynamic threshold properly suppresses
the large fluctuation induced by the cross correlations of individual stock prices and creates a rather robust and stable network structure during the dynamic evolution, in comparison to the static threshold.
Long-range time correlations are revealed for the average clustering coefficient, the average degree and the cross correlation of degrees.