In this paper the device of models copula and
opportunity of application is considered at the analysis of
time series. For the purpose the copula definition and its
properties, are discussed then different copulas families.
The illustrative examples of modeling of distribution by
various kinds copulas are described, using a package R

Novik, A.

Troush, N.

English

Электронная библиотека БГУ

148 
Construction of Copula Models 
A. Novik
1)
, N.N.Troush
2)
 
1). Janka Kupala State University of Grodno, BELARUS, Grodno, str. Limoga 39-48 
NovAnn81@mail.ru 
2). Belarusian State University, 220030 Minsk, Nezavisimosti av. 4, troush@bsu.by 
 
Abstract: In this paper the device of models copula and 
opportunity of application is considered at the analysis of 
time series. For the purpose the copula definition and its 
properties, are discussed then different copulas families. 
The illustrative examples of modeling of distribution by 
various kinds copulas are described, using a package R.  
Keywords: Copula, elliptical, archimedean, extreme. 
1. INTRODUCTION 
The device copulas has received wide application to 
such financial tasks, as an estimation of risk, choice of an 
optimum investment portfolio, estimation of cost of 
difficult financial tools, management of currency risk of 
banks. Copulas already have found successful application 
at modeling the natural phenomena for example, on 
oceanology (Michele С., 2007), at research of ways of 
hashing given as means of protection of a database 
(Sarathy R., 2002), at the analysis of the microeconomic 
data (Smith M., 2003; Genius M., Strazzera E., 2004; Sun 
J., 2008; Kim G., 2007). 
By results of events of the crisis period 2008 - 2009 
Basel Committee on bank supervision has published the 
document “Bank regulation and supervision (BCBS II)”, 
the describing order of an estimation is brave for the 
purpose of account of sufficiency of the capital of banks. 
In the document the preference is given up to models 
“copula” in comparison with the approaches, most 
widespread in bank practice, (summation, simple 
diversification and variance- covariance the approach). It 
is explained to that the models “copula” allow to simulate 
joint multidimensional of distribution (including 
asymmetric), which are not normal. In October, 2010 in 
the document Basel Committee “Development of the 
approaches to modeling aggregation is brave” the basic 
supervision on questions aggregation are generalized is 
brave, allowed to allocate advantages received from 
application of models “copulas”. Thus study of models 
“copula” represents the special urgency in view of the 
legislative initiatives on regulation is brave of commercial 
banks and control of a level of sufficiency of their capital. 
2. DEFINITION OF TWO-DIMENSIONAL 
COPULA AND BASIC PROPERTIES 
Definition. Function ),( yxС  is called subcopula two 
variable x  and y , defined on BA , where 
]1,0[],1,0[  BA , with range space ]1,0[  and satisfy the 
conditions : 
1. 0),0(,0)0,(  yСxС ; 
2. yyСxxС  ),1(,)1,( ; 
3. 0),(),(),(),( 21121122  yxСyxСyxСyxС , 
where 
2
11 ]1,0[),( yx , 
2
22 ]1,0[),( yx  such as 
2211 , yxyx    
Definition. Copula – is a subcopula in cases when 
]1,0[],1,0[  BA .  
Properties of the copula: 
1. Boundedness: 1),(0  yxС . 
2. Any copula lays in borders Frechet-Hoeffding: 
),min(),()0,1max( yxyxCyx  . 
3. Domination. Copula 2C  dominate  above copula 
1C , or 21 CC  , in case, when for yx,  it it true 
),(),( 21 yxCyxC  . 
4. For 
2
21
2
21 ]1,0[),(,]1,0[),(  yyxx  it is true 
12121122 ),(),( yyxxyxСyxС  . 
Use copulas for modeling joint probability of 
distributions is based on conclusions of the theorem Sklar, 
entered in 1959.  
Sklar’s theorem. Let Ryx , random variables, 
marginal distributions functions )()( yYPyFY   
),()( xXPxFX   two-dimesional the function of 
distribution looks ),(),( yYxXPyxFXY  . Then 
exists ),( yxC , such as: 
))(),((),( yFxFCyxF YXXY  .  (1) 
If functions )(xFX  и )(yFY  are continuous, then 
copula (1) is unique, otherwise, copula ),( yxC  can be 
always determined on area of meanings of functions  
)(xFX  and )(yFY . Correctly and return statement, if 
),( yxC  - copula, )(xFX  and )(yFY  - marginal 
distributions functions, then function ),( yxFXY , 
determined above, is two-dimensional function of 
distribution of a random vector (X, Y). 
Copula is probability of approach of joint event for 
random variables Ryx , , that is  
))(),((),( 11 yFxFFyxC YXXY
 . 
Thus, copula – is function, allowing to proceed from one-
dimensional distributions of random variables to joint 
distribution.  
3. RESEARCH OF VARIOUS FAMILIES 
COPULAS 
All copulas can be related to three families: elliptical 
(Gaussian, Student’s), archimedean copula (Frank, 
Clayton, Gumbel, Ali-Mikhail-Haq), extreme-value 
(Galambos, Husler-Reiss).  
Before to proceed to consideration of separate 
families copulas, is necessary to enter such concepts, as 
product, maximal and minimal copulas. 
Copula ),( yxC
 
for two variables X, Y is called 
product copula, if
 
xyyxC ),( . 
149 
  
Fig. 1- The product copula and its level curves. 
The minimum copula- it is the bottom border for all 
copul, has the following definition: 
)0,1max(),(  yxyxC
 
  (2) 
 
Fig. 2- The minimum copula and its level curves. 
The maximum copula- is the top border for all copul, 
has the following definition: 
),min(),( yxyxC    (3) 
 
Fig. 3- The maximum copula and its level curves. 
For construction of the basic families copulas 
environment R, intended for automation statistical of 
calculations was used. The examples of generation of two 
joint distributions with the different forms of interrelation 
are resulted. 
Definition.  Elliptical copula satisfies to the following 
formula:  
))(),((),( 11 yFxFFyxC YXXY
 ,  (4) 
где )(),(),,( yFxFyxF YXXY  functions of distribution of 
the normal laws. 
Exist Gaussian and Student's elliptical copulas. They 
occur from the analytical forms of record two-
dimensional of normal distribution and distribution 
student's.  
Example 1. The Gaussian copula is defined as follows: 
212
2
2
2
121
)( )(
2 )1(2
2
exp
12
1
),(
1 1
dzdz
zzzz
yxС
xF yFX Y











  
 
 



(5) 
where  1,1 -parameter of copula. 
Is investigated Gaussian copula at various meanings of 
parameter θ. With the help of a statistical package R the 
construction copulas and contour diagram. 
 
Fig. 4- The Gaussian copula and its level curves, with θ=-1. 
 
Fig. 5- The Gaussian copula and its level curves, with θ=-1. 
Example 2. Student’s copula  is defined as follows: 
21
2
2
2
21
2
2
2
1
)( )(
2 )1(
2
1exp
12
1
),(
1 1
dzdz
zzzz
yxС
xF yFX Y


 












  
 




(6) 
where  1,1 -parameter of copula , ν- number of 
degrees of freedom copula. 
Is submitted Student’s copula at various value of 
parameter θ. 
 
Fig. 6-The Student’s copula and its contour diagram, if θ=-1. 
 
Fig. 7- The Student’s copula and its contour diagram, if θ=1. 
Archimedean copulas may be constructed using a 
function ],0[]1,0[:   is called a generator and has 
the following properties: 
1.  )0(,0)1(   
2. Continuous; 
3. Decreasing: )1,0(,0)(  tt   
4. Convex: )1,0(,0)(  tt  
The pseudo-inverse of φ(t) must also be defined, as 
150 
follows: 








tt
tt
)(,0
)0(0),(1]1[


  
and tt  ))((]1[  , for every ]1,0[t . 
Definition. Archimedean copula is generated as follows: 
)]()([),( 1 yxyxC    , 
Basic archimedean copula the following are: Frank, 
Clayton, Gumbel, Ali-Mikhail-Haq.  
Example 3. The Ali-Mikhail-Haq copula is defined as 
follows: 
)1)(1(1
),(
yx
xy
yxС



  (7) 
where  1,1 – parameter of copula, describing a 
degree of dependence of random variables. To be under 
construction Ali-Mikhail-Haq copula in a statistical 
package R for various value of parameter θ 
 
Fig. 8- The Ali-Mikhail-Haq copula and its contour diagram, 
if θ=-1. 
   
Fig. 9- The Ali-Mikhail-Haq copula and its contour diagram, 
if θ=1. 
Example 4. The copula Clayton is defined as follows: 
 












 1)1()1(1),(
11
yxyxyxC    (8) 
where ),0()0,1[   
In a package R is under construction Clayton copula 
and its contour diagram at various value of parameter θ. 
  
Fig. 10- The Clayton copula and its level curves, with θ=-1. 
 
Fig. 11- The Clayton copula and its level curves, if θ=5. 
Example 5. The Frank copula is defined as follows 
 











1
)1)(1(
1ln
1
),(


 e
ee
yxC
yx
,
 (9) 
where ),0()0,(  . 
Is submitted Frank copula at various value of parameter θ.  
  
Fig. 12- The Frank copula and its contour diagram, if 
θ=0,02. 
 
Fig. 13- The Frank copula and its contour diagram, if θ=50. 
Example 6. The Gumbel copula is defined as follows: 
   1)ln()ln(exp),( yxyxC 
 (10)
 where ),1[  . 
In a statistical package R is constructed Gumbel 
copula with the contour diagram for various value of 
parameter θ. 
  
Fig. 14- The Gumbel copula and its contour diagram 
projection on an axis x,y, if θ=1. 
151 
 
Fig. 15- The Gumbel copula and its contour diagram 
projection on an axis x,y, if θ=10. 
Definition. Extreme-value copula is generated as 
follows: 
0),,(),(  tyxCyxC ttt .(11) 
Basic extreme-value copula the following are 
Galambos, Husler-Reiss. 
Example 7. The Galambos copula is defined as follows: 
   1)ln()ln(exp),(   yxxyyxC  (12) 
where ),0[  . 
Is submitted Galambos copula at various value of 
parameter θ. 
 
Fig. 16- The Galambos copula and its contour diagram 
projection on an axis x,y, if θ=0,01. 
 
Fig. 17- The Galambos copula and its contour diagram 
projection on an axis x,y, if θ=10. 
Example 8. The Husler-Reiss copula is defined as 
follows: 



















x
y
y
y
x
xyxC
ln
ln
ln
2
1
ln
ln
ln
2
1
exp),(




 
(13) 
where ),0[  . 
In a statistical package R is constructed Husler-Reiss 
copula with the contour diagram projection on an axis x,y 
for various value of parameter θ. 
 
Fig. 18- The Husler-Reiss copula and its level curves, with 
θ=0,01. 
  
Fig. 19- The Husler-Reiss copula and its level curves, with 
θ=20. 
4. CONCLUSION 
In job is given the definition of copula and its 
properties. The families copulas are investigated, such as 
elliptical, archimedean, extreme-value. In a statistical 
package R the following kinds copulas are constructed: 
Gaussian, Student’s, Frank, Clayton, Gumbel, Ali-
Mikhail-Haq, Galambos, Husler-Reiss for various value 
of parameter θ. The device of copulas allows, knowing 
marginal of distribution to proceed to joint distribution of 
two random variables. The copula contains the 
information on a nature of dependence between two 
random variables, which no in marginals distributions, but 
does not contain the information about marginals 
distributions. 
5. REFERENCES 
[1] U. Cherubini, E. Luciano, W. Vecchiato. Copula 
Methods in Finance. John Wiley & Sons Ltd. 
2004.p.293. 
[2] R. Nelsen. An Introduction to Copulas. Second 
Edition. Springer. New York. 2006. p. 269. 
 
 


Construction of Copula Models

http://elib.bsu.by/bitstream/123456789/9322/1/34r%20148.pdf

Construction of Copula Models

Abstract

Similar works

Full text

Available Versions

Электронная библиотека БГУ