Intelligent Information Ma nagement, 2011, 3, 137-141
doi: 10.4236/iim.2011.34017 Published Online July 2011 (http://www.SciRP.org/journal/iim)
Copyright © 2011 SciRes. IIM
A Dynamic Cross Contagion Model of Curr ency Crisis
between Two Countries*
Yirong Ying1, Xiangqing Zou1, Ke Chen1,2, Yuyuan Tong1
1 Shanghai University, Shanghai, China
2 Chongqing Jiaotong University, Chongqing, China
E-mail: ckbest@163.com
Received March 31, 2011; revised April 26, 2011; accepted May 10, 2011
Abstract
The contagion aspect of the currency crisis is an important research issue today. In this paper, we set up a
dynamic differential model of currency crisis cross contagions between two countries by expanding general-
ized logistics model, and analyze all kinds of possible equilibrium conditions. It is probably a new idea of
studying currency crisis contagion mechanism.
Keywords: Cross Contagion, Currency Crisis, Differential Dynamic Model
1. Introduction
Since 1970s, currency crisis have occurred frequently on
a global scale. Especially, in the past twenty years, there
have happened several currency crises with significant
influence: European exchange rate system crisis in
1992-1993, Mexican crisis in 1994-1995, southeast
Asian crisis in 1997-1998, Russian rubles crisis in 1998,
Brazilian curren cy crisis in 1998-1999, Turkish lire crisis
in 2000-2001, Argentine peso crisis in 2001-2002 and the
global crisis triggered by American subprime crisis in
2007. They may be caused by unreasonable domestic
economic structures, heavy debt, unsuccessful monetary
policy or external shocks caused by international finan-
cial speculators. However the worse thing than crises
happening frequ ently is crisis contagions within a region
or global area. This leads to more crises impacts on the
economic or financial system over the world. And the
crisis contagions tend to be stronger and stronger, wider
and wider. From the phenomenon perspective, the simple
reason of stronger and wider crises contagions is that the
continuous development of international trade and capi-
tal flows make closer and closer relations of global
economy and finance. But beyond the simple reason,
what is complex mechanism of monetary crisis contag ion?
It will be the focus of this paper.
Since three generations of models were developed to
explain financial crises, many scholars have done a lot in
the area of currency crisis contagion mechanism and con-
tagion path. Olivier Loisel and Philippe Martin (2001)
presented a micro-fou nded model where governments had
an incentive to devalue to increase the national market
share in a monopolistically competitive sector. The con-
clusion showed that the more important trade competition,
the more likely self-fulfilling speculative crises and the
larger the set of multiple equilibria. They also concluded
that coordination decreased the possibility of simultane-
ous self-fulfilling speculative crises in the region and re-
duced the set of multiple equilibria. However, regional
coordination, even though welfare improving, makes
countries more dependent on other countries’ fundamen-
tals so that it may induce more contagion [1]. Helmut Stix
(2007) studied the effects of France interventions during
the 1992-1993 European Monetary System crises. In his
paper, a Markov Switching model is estimated where
interventions influence the probabilities of transition be-
tween one calm and turbulent regime [2]. Eelke de Jong,
Willem F. C, Verschoor and Remco C. J. Zw inkels (2009)
estimated a dynamic hetero-generous agent model for the
British pound during the European monetary system crisis
and illustrate the ch ain of events leading to the suspension
of the pound from the exchange rate mechanism in terms
of switching beliefs [3]. Li Gang, Pan Hao-min and Jia
Wei (2009) empirically studied spatial convergence and
contagious pathes of financial crisis using methods of
spatial statistical analysis. They concluded that conta-
gious paths o f subprime crisis in the United States are the
location of geographic regions, G7 political groups, trade
*This work has been supported by the Research Fund of Program Found-
ation of Education Ministry of China (10YJA790233).
Y. R. YING ET AL.
Copyright © 2011 SciRes. IIM
138
relations and the openness of capital items. Contagious
pathes of Southeast Asia’s financial crisis a re the location
of geographic regions, trade relations and the openness of
capital items [4]. Juan Gabriel Brida, David Matesanz
Gómez and Wiston Adrián Risso (2009) introduced a new
method to describe dynamical patterns of the real ex-
change rate co-movements time series and to analyze
contagion in currency crisis. The method combines the
tools of symbolic time series analysis with the nearest
neighbor single linkage clustering algorithm. By data
symbolizing, they obtained a metric distance between two
different time series that was used to construct an ul-
tra-metric distance. By analyzing the data of various coun-
tries, they derived a hierarchical organization, constructing
minimal-spanning and hierarchical trees. From these trees
they detected different clusters of countries according to
their proximity. They concluded that the methodology
permits them to construct a structural and dynamic topol-
ogy that was useful to study interdependence and conta-
gion effects among financial tim e series [5].
2. The Dynamic Cross Contagion Model
between Two Countries
Logistic models are widely applied to many research
areas such as biological, economics, sociology. Ying and
Zou (2010) set up a currency crisis infectious differential
dynamic model between two countries by expanding
generalized logistics model as following [6].
111
222
d1
d
d1
d
xx
x
tky
yy
y
tkx



 




(1)
The system (1) can be easily deduced to a general
form.
22
1
112
11
2
221
d
d
d
d
k
x
y
xx
tky
k
x
y
yy
tkx


(2)
In this paper, we consider a more general dynamic
model of financial crisis cross contagion between two
countries.
22
1
111 2
11
2
221 2
d
d
d
d
k
x
y
xx
tkxy
k
x
y
yy
tkxy





(3)
where
x
,ystand for exchange rate of country A and
exchange rate of country B respectively, 1
,2
the in-
trinsic growth of exchang e rate of country A and country
B respectively, 1
k,2
k the superior limit of exchange
rate changing of country A and country B respectively,
1
the crisis infect coefficient B to A, 2
the crisis infect
coefficient A to B, 1
the “addicted-to-absorption”
coefficient of country A toward the crisis infect, 2
“addi cted-to-absorp tion” coefficient of country B toward
the crisis infect, 01
i
,0
i
,0
i
k and 1
i

(1, 2i
).
In order to be convenient, we transform (3) to (4).


1
2
d,
d
d,
d
x
x
Fxy
t
yyFx y
t
(4)
where

22
1
11
11 2
,
k
x
y
Fxy kxy

 (5)

11
2
22
21 2
,
k
x
y
Fxy kxy

 (6)
The parameters i
i
i
k (1, 2i) ar e all positive
constants and 1
i
 (1, 2i
).
We denote the region (see Figure 1)
21
12
(, )0,0
kk
Dxy xy

(7)
3. Stability Analysis
Proposition 1: If 1
i
 (1, 2i),01
i
,
2
112 2
kk

,2
22 11
kk
, then there exists an unique
equilibrium point
,Qxyin the region D, and it is a
stable node or focus.
Proof: After a simple calculation, we can obtain an
equil-brium point
,Qxyin addition to another three
equilbrium points
0,0 ,
1
12
,0
k




2
12
0, k



and

,Qxy
where

2
221 1
12 12
1
kk
x



2
112 2
12 12
1
kk
y


(8)
If it satisfies the conditions of proposition 1, then we can
easily obtain
Y. R. YING ET AL.
Copyright © 2011 SciRes. IIM
139
Figure 1. The equilibrium point Qin the region D.
1
2
0k
x
 2
1
0k
y
 (9)
We can conclude that there is a unique equilibrium
point in regionD.
By using Taylor formula, we can transform (4) to a
form as following,
 
 
11
22
d
d
d
d
QQ
QQ
FF
x
x
xx yy
tx y
FF
y
y
xx yy
tx y




 






(10)
where Q point of coordinates are

1Q
x
xxx
 ,

2Q
y
yyy
 (11)
and 01
i


1, 2i.
The equivalent form for system (10) is








11 12
21 22
d,,
d
d[,, ]
d
QQ QQ
QQ QQ
x
x
axyxx axyyy
t
yyaxyx xaxyyy
t




(12)
After simple calculating, we can obtain



2
2212 1
1
11 12
11 2
2
,
k
x
ykyx
axy kxy



 


 (13-1)
 
211
1
121 22
11 2
(1 )
,
k
x
kx
axy kxy
 




 (13-2)



122
2
212 12
21 2
1
,
k
yk y
axykxy
 




 (13-3)



2
11212
2
22 22
21 2
2
,
k
x
yk xy
axy kxy



 


 (13-4)
Embedding the coordinates of point

,Qxy into
equations from (13-1) to (13-4) one by one, we can ob-
tain expression as following respectively.


2
11 12
11 2
,
QQ
x
axy A
 
 (14-1)


2
121 1
1
12 2
1
,
QQ
k
x
kx
axy A
 

(14-2)


1
212 2
2
21 2
1
,
QQ
k
yk y
axy B
 

(14-3)


2
22 12
22 2
,
QQ
y
axyB
 
 (14-4)
where


22
12 221 1
12 12
1
A
kk
 


(15)


22
21 112 2
12 12
1
Bkk
 


(16)
In order to judge the state of the equilibrium pointQ,
we consider the characteristic equation of equation (4) at
the pointQ.
20pq

 (17)
where
 
11 22
,,
QQ QQ
paxyaxy
 
(18)
 
11 221221
,, ,,
QQQQQQ QQ
qaxyaxya xya xy
(19)
Y. R. YING ET AL.
Copyright © 2011 SciRes. IIM
140
We can judge the state of the equilibrium point Q
according ODE theory when we embed (14-1), (14-2),
(14-3) and (1 4-4) into equation (18) one by one.
Proposition 2: If12 12
min{ ,}0


, then equilib-
rium point Q is a stable node or focus.
ProofEmbed14-1 and 14-4into inequality as
following

11 22
,,0
QQ QQ
axy axy

 

(20)
After calculating, we can easily obtain




1 1122212
22
22
12 21
0
 
 



(21)
So the proposition 2 is obviously true.
Proposition 3: If12 12
min{ ,}0

, then equilib-
rium point Q is an unstable node or focus.
Proof: Embed14 -1and 14-4into inequality as
fol-lowing

11 22
,,0
QQ QQ
axy axy



(22)
After calculating, we can easily obtain



1 1122212
22
22
12 21
0
 
 



(23)
So the proposition 3 is obviously true.
Proposition 4: There exists at least a pair of1
and 2
Which satisfies conditions

1122120
 
 (24)
and such that 0p. In this situation, problem of judg-
ing center or focus appears.
Proof: We can find a pair of 1
and2
as following


22
2
11212
12
22
212 1
 



 
(25)
which satisfies condition

112 2120
 

and such that 0p.
So the proposition 4 is true.
Proposition 5: If0q, then equilibrium point Q is
unstable saddle point.
Proof: Embed (14-1), (14-2), (14-3) and (14-4) into
(19), we can obtain

22
1212121 20
 
 (25)
Here 1
2
are both real numbers. Thusif (25) is
true1
2
must satisfy the condition as following

2
12 12
40
 
 (26)
However, In fact, 1
and 2
can not meet inequality
(26) in any case. So the proposition 5 is true.
4. Conclusions
In nature some vegetables tend to absorb pollutants like
phenol. This is known as “addicted-to-absorption” phe-
nomenon. Alternatively, some plants have a pronounced
tendency for “anti-absorption.” Financial markets exhibit
similar characteristics. When a country is in financial
crisis, financial panic ensues and the investors withdraw
their money from neighbor or other countries. This
makes the financial crisis spread to other countries, and
accelerates the decline of prices or the return on assets.
This is called the “herding effect.” If the affected coun-
tries, however, give investors confidence in a timely
manner or show stronger economical and financial sup-
port, panic among investors can be alleviated. This
makes investors to bring more funds to the affected
country making asset prices and yields to rise.
Therefore, based on Ying & Zou’s (2010) contagion
model, we introduce a factor 1
x
or 2y
to reflect such
an accelerating phenomenon for better describing the
dynamic behavior of financial contagion between the two
countries. Inspired by the “addicted-to-absorption” phe-
nomenon of the plant, we defined i
as the “ad-
dicted-to-absorption” coefficient of icountry’s financial
markets reflecting to financial contagion. And we define:
if 01
i
, the changes of exchange rate in country
iwill accelerate its volatility; 0
i
States that the ex-
change rate changes will decrease its volatility.
Furthermore, after analyzing the stability of the pro-
moting model by Ordinary Differential Equation Qualita-
tive Theory, we find: under the assumption that the upper
limit of exchange rate change in a country depends on the
ratio between the fluctuation limit of another country’s
exchange rate and its transmission coefficients, there ex-
ists some linear combinations of i
and i
to reflect
that the financial crisis contagions are becoming uncon-
trollable or controllable. At some angle, the finding is
closer to the actual state of the financial contagions. For
further study, we will find the approximate calculation
method of i
and i
by discretizing the promoting dif-
ferential dynamic model. It is a simple way to do empiri-
cal analysis of financial contagions between two countries.
This will be our main task of the future.
5. References
[1] O. Loisel and P. Martin, “Coordination, Cooperation,
Contagion and Currency Crises,” Journal of International
Economics, Vol. 53, No. 2, 2001, pp. 399-419.
doi:10.1016/S0022-1996(00)00055-6
[2] H. Stix, “Impact of Central Bank Intervention during
Periods of Speculative Pressure: Evidence from the Euro-
Y. R. YING ET AL.
Copyright © 2011 SciRes. IIM
141
pean Monetary System,” German Economic Review, Vol.
8, No. 3, 2007, pp. 399-427.
doi:10.1111/j.1468-0475.2007.00412.x
[3] E. de Jong, W. Verschoor and R. Zwinkels, “A Hetero-
geneous Route to the European Monetary System Crisis,”
Applied Economics Letters, Vol. 16, No. 9, 2009, pp. 929-
932. doi:10.1080/13504850701222152
[4] G. Li, H. M. Pan and W. Jia “Statistics for Microarrays:
Design, Analysis,” Statistical Research, Vol. 26, No. 12,
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[5] J. G. Brida, D. M. Gómez and W. A. Risso, “Symbolic
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tion to Contagion in Currency Crises,” Expert Systems
with Applications, Vol. 36, No. 4, 2009, pp. 7721-7728.
doi:10.1016/i.eswa.2008.09.038
[6] Y. R. Ying and X. Q. Zou, “Study on a Contagion Model
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