Financial Networks with Static and

dynamic thresholds

Tian QiuNanchang Hangkong University

2

Outline

Motivation Financial networks with static and dy

namic thresholds Topology dynamics Economic sectors Conclusions

3

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.

4

Financial networks with static and

dynamic thresholds

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

9

Topology dynamics detrended fluctuation analysis(DFA)

Average clustering coefficient

Average degree

cross correlation of degrees

10

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()(

11

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

12

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

13

Topology dynamics-Average clustering coefficient

T

t

tCT

C1

)(1

0.78

0.68

0.88

0.85

14

Topology dynamics-Average clustering coefficient

15

Topology dynamics-Average degree

where is the degree of node

N

ii tkN

tK1

)(1

)(

The average degree is defined by

i)(tki

16

Topology dynamics-Average degree259N

17

Topology dynamics-Average degree

T

t

tKT

K1

)(1

18

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

19

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

20

Degree distribution

21

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.

22

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.

23

Topology dynamics-cross correlation of degrees

T

t

trT

r1

)(1

0.00

-0.20

0.36

0.22

24

Topology dynamics-cross correlation of degrees

25

Economic sectors

26

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.

27

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

28

Topology dynamics- Economic sectors

sQ1.1 )(6 tQd

29

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.

30

Thank You!