A Dynamic Cross Contagion Model of Currency Crisis between Two Countries

doi:10.4236/iim.2011.34017

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Intelligent Information Ma nagement, 2011, 3, 137-141

doi: 10.4236/iim.2011.34017 Published Online July 2011 (http://www.SciRP.org/journal/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: [email protected]

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.

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

tky

tkx











 













(1)

The system (1) can be easily deduced to a general

form.

112

221

tky

tkx



























(2)

In this paper, we consider a more general dynamic

model of financial crisis cross contagion between two

countries.

111 2

221 2

tkxy































(3)

where

,ystand for exchange rate of country A and

exchange rate of country B respectively, 1



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,



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





,0



,0

k and 1





(1, 2i



In order to be convenient, we transform (3) to (4).



Fxy

yyFx y













(4)

where



11 2

Fxy kxy











 (5)



21 2

Fxy kxy











 (6)

The parameters i



，i



，i

k (1, 2i) ar e all positive

constants and 1



 (1, 2i



We denote the region (see Figure 1)

(, )0,0

Dxy xy

















 (7)

3. Stability Analysis

Proposition 1: If 1



 (1, 2i),01



,

112 2



,2

22 11



, 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 ,









，2

0, k









and



,Qxy

where



221 1

12 12









，



112 2

12 12









 (8)

If it satisfies the conditions of proposition 1, then we can

easily obtain

Y. R. YING ET AL.

139

Figure 1. The equilibrium point Qin the region D.



 ，2



 (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,

 

xx yy

tx y

xx yy

tx y









 















(10)

where Q point of coordinates are



xxx



 ,



yyy



 (11)

and 01







1, 2i.

The equivalent form for system (10) is



11 12

21 22

d,,

d[,, ]

QQ QQ

axyxx axyyy

yyaxyx xaxyyy

















(12)

After simple calculating, we can obtain



2212 1

11 12

11 2

ykyx

axy kxy









 





 (13-1)

 

211

121 22

11 2

(1 )

axy kxy





 









 (13-2)



122

212 12

21 2

yk y

axykxy





 









 (13-3)



11212

22 22

21 2

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.



11 12

11 2

axy A

 



 (14-1)



121 1

12 2

axy A

 

















 (14-2)



212 2

21 2

yk y

axy B

 

















 (14-3)



22 12

22 2

axyB

 



 (14-4)

where



12 221 1

12 12

 







 (15)



21 112 2

12 12

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.

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.

Proof：Embed（14-1 ）and 14-4）into inequality as

following







11 22

,,0

QQ QQ

axy axy



 



(20)

After calculating, we can easily obtain



1 1122212

12 21

 

 







(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: Embed（14 -1）and （14-4）into inequality as

fol-lowing







11 22

,,0

QQ QQ

axy axy







(22)

After calculating, we can easily obtain









1 1122212

12 21

 

 







(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



11212

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



1212121 20

 

 (25)

Here 1



，2



are both real numbers. Thus，if (25) is

true，1



，2



must satisfy the condition as following



12 12

 

 (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



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



, the changes of exchange rate in country

iwill accelerate its volatility; 0



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.

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,

2009, pp. 81-87.

[5] J. G. Brida, D. M. Gómez and W. A. Risso, “Symbolic

Hierarchical Analysis in Currency Markets: an Applica-

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

of Currency Crisis between Two Countries,” Molecular

Biology, Biophysics and Bioengineering of the CMBB,

Qiqihar, China, 2010, pp. 599-602.