CALIBRATING AND SIMULATING COPULA FUNCTIONS: AN APPLICATION TO THE ITALIAN STOCK MARKET

Claudio Romano1

Abstract Copula functions are always more used in financial applications to determine the dependence structure of the asset returns in a portfolio. Empirical evidence has proved the inadequacy of the multinormal distribution, commonly adopted to model the asset return distribution. Copulas are flexible instruments used to build efficient algorithms for a better simulation of this distribution.

The aim of this paper is describing the statistical procedures used to calibrate a copula function to real market data. Then, some methods used to choose which copula better fit data are presented. Finally a number of algorithms to simulate random variate from certain types of copula are illustrated.

The procedures described are applied to a portfolio of Italian equities. We show how to generate efficient Monte Carlo scenarios of equity log-returns in the bivariate case using different copulas.

Keywords: Copula Function, Dependence Structure, Multivariate Distribution Function.

1 Corresponding author: Claudio Romano, Risk Management Function, Capitalia, Viale U. Tupini, 180, 00144 – Rome, Italy. E-mail: claudio.romano@capitalia.it. The author is grateful to Prof. G. Szegö for his valuable comments and suggestions that helped improve the article substantially.

2

CALIBRATING AND SIMULATING COPULA FUNCTIONS: AN APPLICATION TO THE ITALIAN STOCK MARKET

Claudio Romano

Introduction Copula functions are used in financial application since 19992. Empirical evidence has proved that the multinormal distribution is inadequate to model portfolio asset return distribution under two points of view:

1) The empirical marginal distributions are skewed and fat tailed;

2) it does not consider the possibility of extreme joint co-movement of asset returns3. In other words, the dependence structure is different from the Gaussian one.

Copula functions are a useful tool to implement efficient algorithms to simulate asset return distributions in a more realistic way. In fact, they allow to model the dependence structure indipendently from the marginal distributions. In this way, we may construct a multivariate distribution with different margins and the dependence structure given from the copula function.

Therefore, a crucial step is the selection and the calibration of the copula function from real data. In this paper a collection of methods for calibrating, selecting and simulating copula functions are presented. Our aim is to collect in this article the principal contributions to the argument provided by the international literature cited in the references.

Most of the method presented are applied to an empirical data set of the log-returns of two Italian equities. When it is possible, we show as the copula approach performs better than the multinormal distribution in modelling real data.

The rest of this paper is structured as follows. In section one, a brief definition of copula function is given, describing the main families of copula used in practical applications4. In section two, some methods to estimate the parameters of a determined copula function from real data are presented. The

2 See Embrechts, McNeil and Straumann (1999). 3 As in the case of a market crash. 4 i.e.: the class of the elliptical copulas and the class of the Archimedean copulas.

3

procedures to select the type of copula which better fits empirical data are showed in section three. In section four, the algorithms to simulate random variates from some types of copula are reported. An application to a time series of the log-returns of two Italian equities if performed in section five. Finally, we draw some concluding remarks.

1. Definition of copula function An n-dimensional copula5 is a multivariate distribution function (d.f.) , C, with uniform distributed margins in [0,1] (U(0,1)) and the following properties:

1. C: [0,1]n → [0,1];

2. C is grounded and n-increasing;

3. C has margins Ci which satisfy Ci(u) = C(1, ..., 1, u, 1, ..., 1) = u for all u∈ [0,1].

It is obvious, from the above definition, that if F1, ..., Fn are univariate distribution functions, C(F1(x1), ..., Fn(xn)) is a multivariate d.f. with margins F1, ..., Fn, because Ui = Fi(Xi), i = 1, ..., n, is a uniform random variable. Copula functions are a useful tool to construct and simulate multivariate distributions.

The following theorem is known as Sklar’s Theorem. It is the most important theorem about copula functions because it is used in many practical applications.

Theorem6: Let F be an n-dimensional d.f. with continous margins F1, ..., Fn. Then it has the following unique copula representation:

F(x1, …, xn) = C(F1(x1), ..., Fn(xn)) . (1)

From Sklar’s Theorem we see that, for continous multivariate distribution functions, the univariate margins and the multivariate dependence structure can be separated. The dependence structure can be represented by a proper copula function. Moreover, the following corollary is attained from (1).

Corollary: Let F be an n-dimensional d.f. with continous margins F1, ..., Fn and copula C (satisfying (1)). Then, for any u=(u1,…,un) in [0,1]n:

( ))(),...,(),...,( 11

111 nnn uFuFFuuC −−= , (2)

5 The original definition is given by Sklar (1959). 6 For the proof, see Sklar (1996).

4

where Fi-1 is the generalized inverse of Fi.

A trivial example is the copula of independent random variables (the product copula). It takes the form:

nnind uuuuC ⋅⋅= ...),...,( 11 .

Another example is the Farlie-Gumbel-Morgenstern (FGM) copula, which in the bivariate case is defined by

[ ] 11 , )1)(1(1),( 212121 ≤≤−−−+= αα uuuuuuC .

1.1. Elliptical copulas The class of elliptical distributions provides useful examples of multivariate distributions because they share many of the tractable properties of the multivariate normal distribution. Furthermore, they allow to model multivariate extreme events and forms of non-normal dependencies. Elliptical copulas are simply the copulas of elliptical distributions. Simulation from elliptical distributions is easy to perform. Therefore, as a consequence of Sklar’s Theorem7, the simulation of elliptical copulas is also easy.

1.1.1. Normal copula The Gaussian (or normal) copula is the copula of the multivariate normal distribution. In fact, the random vector X=(X1,…,Xn) is multivariate normal iff:

1) the univariate margins F1, …, Fn are Gaussians; 2) the dependence structure among the margins is described by a unique

copula function C (the normal copula) such that8:

( ))(),...,(),...,( 11

11 nRn

GaR uuuuC −−Φ= φφ , (3)

where RΦ is the standard multivariate normal d.f. with linear correlation matrix R and 1−φ is the inverse of the standard univariate Gaussian d.f.

If n=2, expression (3) can be written as:

7 See (1) e (2). 8 As one can easily deduce from (2).

5

∫ ∫− −

∞− ∞−

−+−−

−=

)( )(

212

212

2

2/1212

1 1

)1(22exp

)1(21),(

u vGaR dsdt

RtstRs

RvuC

φ φ

π ,

where R12 is simply the linear correlation coefficient between the two random variables.

1.1.2 t-Student copula The copula of the multivariate t-Student distribution is the t-Student copula. Let X be a vector with an n-variate t-Student distribution with ν degrees of

freedom, mean vector µ (for 1>ν ) and covariance matrix Σ− 2νν (for

2>ν )9. It can be represented in the following way:

ZµXd

Sν+= , (4)

where nR∈µ , S~ 2νχ and the random vector Z~ ),( Σ0nN are independent.

The copula of vector X is the t-Student copula with υ degrees of freedom. It can be analytically represented in the following way:

))(),...,(()( 11

1,, n

nR

tR ututtC −−= νννν u , (5)

where jjiiijijR ΣΣΣ= / for { }nji ,...,1, ∈ and where nRt ,ν denotes the

multivariate d.f. of the random vector S/Yν , where the random variable S~ 2

νχ and the random vector Y10 are independent. νt denotes the margins11 of n

Rt ,ν .

For n=2, the t-Student copula has the following analytic form:

∫ ∫− −

∞− ∞−

+−

−+−+

−=

)( )(2/)2(

212

212

2

2/1212

,

1 1

)1(21

)1(21),(

ut vttR dsdt

RtstRs

RvuC ν ν

ν

ν νπ ,

9 If 2≤ν , then the covariance matrix is not defined. 10 Y has an n-dimensional normal distribution with mean vector 0 and covariance matrix R. 11 All the margins are equally distributed.

6

where R12 is the linear correlation coefficient of the bivariate t-Student distribution with ν degrees of freedom, if 2>ν .

1.2. Archimedean copulas An Archimedean copula can be written in the following form:

[ ])(...)(),...,( 11

1 nn uuuuC ψψψ ++= − (6)

for all 1,...,0 1 ≤≤ nuu and where ψ is a function often called the generator, satisfying:

(i) 0)1( =ψ ;

(ii) for all 0)(' ),1,0( <∈ tt ψ , i.e. ψ is decreasing;

(iii) for all 0)('' ),1,0( ≥∈ tt ψ , i.e. ψ is convex.

Examples of bivariate Archimedean copulas are the following: - Product copula

tt ln)( −=ψ ; 2121 ),( uuuuC ⋅= .

- Clayton copula12

0 ,1)( >−= − αψ αtt ; ( ) ααα /12121 1),( −−− −+= uuuuC .

- Gumbel copula13

( ) 1 ,ln)( ≥−= αψ αtt ; [ ]{ }ααα /12121 )ln()ln(exp),( uuuuC −+−−= .

- Frank copula14

R∈−−−= −

−

αψ α

α

,11ln)(

eet

t

;

−−−+−= −

−−

1)1)(1(1ln1),(

21

21 α

αα

α eeeuuC

uu

.

Extensions to the multivariate case are the following:

- Cook-Johnson copula15

12 Clayton (1978), Cook-Johnson (1981), Oakes (1982). 13 Gumbel (1960), Hougaard (1986). 14 Frank (1979). 15 It is a multivariate extension of the Clayton copula.

7

αα

/1

11 1),...,(

−

=

−

+−= ∑

n

jjn nuuuC .

- Gumbel-Hougaard copula

( ) ( ) ( )[ ]

−++−+−−= αααα1

211 ln...lnlnexp),...,( nn uuuuuC .

- Frank copula

( ) ( ) ( )( )

−−⋅⋅−⋅−+−= −−

−−−

111

1...111ln1),...,(21

n

uuu

ne

eeeuuCn

α

ααα

α .

2. Parameter estimation of a given copula

2.1. The Maximum Likelihood (ML) method Let f be the density of the joint distribution F:

∏=

=n

iiinnn xfxFxFcxxf

1111 )())(),...,((),...,(

where fi is the univariate density of the marginal distribution Fi and c is the density of the copula given by the following expression:

n

nn uu

uuCuuc∂∂

∂=...

),...,(),...,(1

11 .

We suppose to have a set of T empirical data of n financial asset log-returns, { } T

ttn

t xx 11 ),...,( ==χ . Let α),,...,( n1 ϑϑϑ = be the parameter vector to estimate, where iϑ , i=1, ...,n is the vector of parameters of the marginal distribution Fi and α is the vector of the copula parameters. The log-likelihood function is the following:

∑∑∑= ==

+=T

t

n

i

tii

T

t

tnn

t xfxFxFcl1 11

11 );(ln));;(),...,;((ln)( in1 α ϑϑϑϑ . (7)

8

The ML estimator ϑ̂ of the parameter vector ϑ is the one which maximize (7), i.e.:

)(maxargˆ ϑϑ l= .

2.2. The method of Inference Functions for Margins (IFM) According to the IFM method16, the parameters of the marginal distributions are estimated separately from the parameters of the copula. In other words, the estimation process is divided into the following two steps:

(i) estimating the parameters iϑ , i=1,...,n of the marginal distributions Fi using the ML method:

∑=

==T

t

tii

i xfl1

);(lnmaxarg)(maxargˆiii ϑϑϑ

where li is the log-likelihood function of the marginal distribution Fi;

(ii) estimating the copula parameters α , given the estimations performed in step (i):

( )∑=

==T

t

tnn

c );;xF;xFc(l1

1ˆ(),...,ˆ(lnmaxarg)maxargˆ ααα n1

t1 ϑϑ

where lc is the log-likelihood function of the copula.

2.3. The Canonical Maximum Likelihood (CML) method The CML method differs from the IFL method because no assumptions are made about the parametric form of the marginal distributions. The estimation process is performed into two steps:

(i) transforming the dataset ),...,( 1tn

t xx , t=1, ..., T, into uniform variates )ˆ,...,ˆ( 1

tn

t uu , using the empirical distributions17;

(ii) estimating the copula parametes as follows: 16 Joe and Xu (1996). 17 In other words, as we will see in section 4.6, the variates )ˆ,...,ˆ( 1

tn

t uu are generated from the empirical copula.

9

∑=

=T

t

tn

t uuc1

1 );ˆ,...,ˆ(lnmaxargˆ αα .

For example, we can estimate the parameter R of the Gaussian copula (3) with the CML or the IFM method in the following way18:

∑=

Τ=T

tCMLIFM T 1

/1ˆ

tt ςςR

where ( ))(),...,( 11

1 tn

t uu −− ΦΦ=tς . In this notation ti

ti uu ˆ= when we are using the

CML method and )ˆ;( iϑtii

ti xFu = when we are using the IFM method, i=1,...,n.

The following recursive procedure19 is used to estimate the parameter R of the tν-Student copula (5):

(i) let 1R̂ be the IFM/CML estimator of the R parameter for the Gaussian copula;

(ii) ∑= Τ−

+

+

+=T

t

nT 1 ˆ11

1ˆ

t1

mt

tΤt

1m

ςRς

ςςR

νν

ν , m=1,2,...,

where ( ))(),...,( 11

1 tn

t utut −−= ννtς ;

(iii) step (ii) is repeated until m1m RR ˆˆ =+ . So, the IFM/CML estimator of the parameter R for the tν-Student copula is ∞= RR CMLIFM

ˆˆ/ .

Mashal and Zeevi (2002) suggest to use the following algorithm to estimate the parameters ν and R of the tν-Student copula:

(i) transforming the dataset ),...,( 1tn

t xx , t=1, ..., T, into uniform variates )ˆ,...,ˆ( 1

tn

t uu , using the empirical marginal distributions.

(ii) Estimate R̂ using the Kendall’s τ non parametric estimator:

= ijijR τπ ˆ2

sinˆ , i,j=1,...,n.

18 See Durrleman, Nikeghbali and Roncalli (2000). 19 See Bouyé et al. (2000).

10

(iii) Perform a numerical search for ν̂ , i.e.,

( ]( )

= ∑=∞∈

T

t

tn

t uuc1

1,2

ˆ,;,...,(logmaxargˆ Rννν

, where

[ ][ ] ∏

=

+−

+−−−

++Γ

+Γ+Γ= n

ii

n

nn

n

y

nuuc

1

2/)1(22/1

2/)(11

1

)/1()2/)1((

)'1()2/(2/)((),;,...,(ν

ν

νν

νννR

yRyR and

))(),...,((),...,( 11

11 nn ututyy −−== ννy .

2.4. Parameter estimation and dependence measures This method works only with one-parameter bivariate copulas. The main dependence measures20 can be written as a function of the copula21. In some cases analytical solutions are available and the copula parameter can simply be written as a function of the dependence measure. Otherwise, numerical procedure are necessary.

For instance, for the Gaussian copula we obtain:

= SR ρπ6

sin212 and

= τπ2

sin12R .

For the Clayton copula:

ττα−

=12 .

For the Gumbel copula:

( ) 11 −−= τα .

For the Morgenstern copula:

Sρα 3= and τα29= .

20 i.e. the rank correlation coefficients: the Spearman’s rho, ρS, and the Kendall’s tau, τ. 21 e.g., see Nelsen (1998).

11

2.5. Non parametric estimation So far, the parameters of a given type of copula are been estimated. Now the empirical copula (or the Deheuvels copula22) is constructed from the sample data. This is any copulas of the empirical multivariate distribution.

Let { })()(1 ,..., t

nt xx be the order statistics and { }tnt rr ,...,1 be the rank statistics,

t=1,...,T of the dataset. We have: ti

ri xx

ti =)( , i=1,...,n.

Any function

[ ]∑∏= =

≤=

T

t

n

itr

ni

tiTT

tTtC

1 1

1 1,...,ˆ 1 (8)

defined on the lattice

=≤≤

= TtniTt

Tt

in ,...,0;1:,...,1

� is an empirical

copula.

The empirical copula density23 has the following expression:

∑ ∑= =

++

+−+−−=

2

1

2

1

11...1

1

11,...,1ˆ)1(...,...,ˆ

i i

nniin

n

n

Tit

TitC

Tt

Ttc .

3. Selecting the right copula In section two, some methods to calibrate the parameters of a given analytical representation of copula function are illustrated. Now the issue is selecting the type of copula which fits better the empirical data.

3.1. Selecting an Archimedean copula The method described in this section24 is able to select the Archimedean copula which fits better real data. An Archimedean copula has the analytical representation given by equation (6). So, in order to select the copula, it is sufficient to identify the generator, ψ .

22 It was introduced in Deheuvels (1979). 23 See Nelsen (1998). 24 The method was developed by Genest and Rivest (1993).

12

In the bivariate case (n=2), Genest and Rivest defined a univariate function, K, which is related to the generator of the Archimedean copula through the following expression:

)(')()(

zzzzK

ψψ

ψ −= . (9)

A non parametric estimation of (9) is the following:

[ ]∑=

≤=T

tztT

zK1

1)(ˆϑ1 (10)

where [ ]∑=

<<−=

T

txxxxi itit

T 1, 22111

1 1ϑ , i=1,...,T.

We choose a parametric representation for the generator25, ψ . Then, the parameter, α of the selected Archimedean copula is estimated using, for istance, the following estimation of the Kendall’s τ 26:

( ) ( )[ ]∑<

−

−⋅−

=

ji

jiji xxxxsignT

2211

1

2τ .

The parameter α may also be estimated using the IFM or the CML method27. Using α , a parametric estimation of (9) is easily obtained.

All the steps described above are repeated for different choices of ψ . In order to select the Archimedean copula which fits better the dataset, Frees and Valdez (1998) propose to use a QQ-plot between (9) and (10).

The optimal copula may also be selected by minimizing the distance based on the L2 norm between (9) and (10)28:

[ ]∫ −=1

0

2

2 )(ˆ)(),ˆ( dzzKzKKKd .

The method described in this section may also be used to graphically estimate the parameter α of a given Archimedean copula.

25 See section 1.2. 26 The method is descripted in section 2.4. 27 See sections 2.2 and 2.3. 28 See Durrleman, Nikeghbali and Roncalli (2000).

13

3.2. Selecting the right copula using the empirical copula Let { } KkkC ≤≤1 be the set of the available copulas. We choose the copula Ck which minimize the following distance, based on the discrete Ln norm, between the same Ck and the empirical copula as defined in (8):

( )2/1

1 1

211

1

,...,,...,ˆ...,ˆ

−

= ∑ ∑= =

T

t

T

t

nk

nkn

nTt

TtC

Tt

TtCCCd . (11)

The distance (11) may also be used to estimate the vector of parameters Θ∈ϑ of a given copula );( ϑuC in the following way:

[ ]2/1

2);()(ˆminargˆ

−= ∑∈Θ∈ �u

uu ϑϑϑ

CC .

4. Simulation algorithms In this section, we show a collection of algorithms to simulate random variates (u1,...,un) from certain types of copula C. For the definition of copula, these random variates ui are determination of correlated uniform(0,1) distributed random variables. So, in order to simulate random variates (x1,...,xn) from a multivariate distribution F with given margins Fi, i=1,...,n, and copula C, we have to invert each ui using the marginal distributions: n1,...,i ),(1 == −

iii uFx .

4.1. Simulation from the Gaussian copula To generate random variates from the Gaussian copula (3), we can use the following procedure. If the matrix R is positive definite, then there are some

nn× matrix A such as R=AAT. It is also assumed that the random variables Z1, ..., Zn are independent standard normal. Then, the random vector Zµ A+ (where Z=(Z1,…,Zn)T and the vector nR∈µ ) is multinormally distributed with mean vector µ and covariance matrix R.

The matrix A can be easily determined with the Cholesky decomposition of R. This decomposition is the unique lower-triangular matrix L such as LLT=R. Hence, one can generate random variates from the n-dimensional Gaussian copula running the following algorithm:

14

� find the Cholesky decomposition A of the matrix R;

� simulate n independent standard normal random variates z=(z1,…,zn)T;

� set x=Az;

� determine the components nixu ii ,...,1 , )( ==φ ;

� the vector (u1, …, un)T is a random variate from the n-dimensional Gaussian copula, Ga

RC .

4.2. Simulation from the tνννν-Student copula

To simulate random variates from the t-Student copula (5), tRC ,ν ,we can use the

following algorithm, which is based on equation (4):

� find the Cholesky decomposition, A, of R;

� simulate n independent random variates z=(z1,…,zn)T from the standard normal distribution;

� simulate a random variate, s, from 2νχ distribution, independent of z;

� determine the vector y=Az;

� set yxsν= ;

� determine the components nixtu ii ,...,1 , )( == ν ;

� the resultant vector is: (u1,…,un)T ~ tRC ,ν .

4.3. Simulation from the Cook-Johnson copula This algorithm is a particular case of the one suggested by Marshall and Olkin (1988) for the generation of multivariate outcomes from a compound copula. To generate random variates from the Cook-Johnson copula with parameter α , we have to perform the steps below:

15

� generate n independent random variates, y1,...,yn from the exponential distribution29 with parameter 1=λ ;

� generate a random variate, z, from a Gamma )1,/1( α distribution independent of y1,...,yn;

� set ( ) α/1/1 −+= zyu jj , j=1,...,n;

� the vector u=(u1,...,un) is generated from the Cook-Johnson copula.

The Cook-Johnson copula reproduces a positive dependence structure. A negative dependence structure may be obtained for same of the variables setting ii uu −=1* .

4.4. Simulation from the Morgenstern copula The following algorithm30 generates bivariate random variates from the Farlie-Gumbel-Morgenstern copula:

- generate independent uniform(0,1) random variates v1 and v2;

- set u1=v1;

- calculate 1)12( 1 −−= uA α and [ ] )12(4)12(1 122

1 −+−−= uvuB αα ;

- set ( ))/2 22 ABvu −= ;

- the vector (u1,u2) is generated from the Farlie-Gumbel-Morgenstern copula.

4.5. A general algorithm to simulate a copula This method is based on the conditional distributions of a random vector U=(U1,...,Un). In the bivariate case, we have:

{ } ),(\Pr 211\21122 uuCuUuU ==≤

where 1

212121

0211\2

),(),(),(lim),(u

uuCu

uuCuuuCuuCu ∂

∂=∆

−∆+=+→∆

.

29 The exponential distribution has the following form: 0 ,1);( >−= − xexF xλλ . 30 Johnson (1987, p.185).

16

The algorithm31 is the following:

� generate two independent uniform(0,1) random variates v1 and v2;

� set u1=v1;

� let C(u2;u1)=C2\1(u1,u2). Set u2=C-1(v2;u1);

� the vector (u1,u2) is generated from the copula C.

For instance, for the bivariate Frank copula, we have:

( )( ) ( )( )111

1),(21

12

211\2 −−+−−= −−−

−−

uu

uu

eeeeeuuC ααα

αα

and

{ } ( )

−+−+−=== −

−−

1)1(11ln1),(:);( 211\221

1ueuu

euuuuCuuuC α

α

α.

The above algorithm may be generalized to the multivariate case:

� generate n independent uniform(0,1) random variates, (v1,...,vn);

� set u1=v1;

� let C(um;u1,...,um-1)=Cm\1,...,m-1(u1,...,um), m=2,...,n, where

{ }

)1,...,1,,...,()1,...,1,,...,(

),...,(),...,(\Pr),...,(

111,...,(

11

),...,(

111111,...,1\

11

11

−−

−

−−−

−

−

∂∂

=

=≤=

mm

uu

mm

uu

mmmmmmm

uuCuuC

uuUUuUuuC

m

m (12)

� Set um=C-1(vm;u1,...,um-1), m=2,...,n;

� The vector (u1,...,un) is generated from the copula C.

This algorithm is computationally intensive for high values of n. In fact, it is a difficult issue to compute the conditional distribution (12).

4.6. Simulation from the empirical copula The below algorithm permits to generate a vector of random variates from the empirical copula (8):

31 Introduced by Genest and MacKay (1986).

17

� randomly draw a complete observation vector ( )tn

t xx ,...,1 from the historical dataset χ ;

� using the empirical distribution functions, iF̂ , to transform each component of the observation vector to a set of uniform variates:

)(ˆ tiii xFu = , i=1,...,n;

� (u1,...,un) is a vector of non-independent uniforms(0,1) that are dependent through the empirical copula.

5. An application to the Italian stock market In this section we apply the methods of calibration and simulation described before. We use a dataset of 1012 daily observations of the log-returns of a group of Italian equities.

We study, for instance, the daily log-returns of the TIM and the Olivetti equities. In Table 1, the principal statistics regarding the above two equities are reported. In Figure 1 we plot the empirical standardized log-returns of TIM against the standardized log-returns of Olivetti.

Table 1: Main statistics of the empirical distribution of the log-returns of TIM and Olivetti.

Mean Standard deviation

TIM 0.000269 0,025799233 Olivetti 0.000919 0,031200767

Linear correlation Spearman’s rho Kendall’s tau

0,522391 0,517832 0,359868

18

Figure 1: Plot of the empirical standardized log-returns TIM/Olivetti.

We have estimated, with the CML method, the parameters of different types of bivariate copula, using the dataset of the 1012 historical daily log-return observations. In this way, we does not consider any particular analytical form for the marginal distributions and only the copula effects are taken into account.

Therefore, we have selected the copula which better approximate the empirical copula using the L2 norm (11). The results are showed in Table 2.

TIM-Olivetti (standardized log-return)

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

TIM (st. log-ret.)

Oliv

etti

(st.

log-

ret.)

19

Table 2: CML estimation of the parameters (α or R12) and calculation of the L2 norm for different copula types.

Copula Parameter estimation 1012/),ˆ(2 CCd

Gaussian 0.53248 0.00451

t5-Student 0.53953 0.00460

t10-Student 0.54037 0.00432

t20-Student 0.53564 0.00446

FGM 1.55349 0.00595

Gumbel 1.56218 0.00839

Frank 3.82211 0.00507

Clayton 1.12436 0.01583

Seeing the results in Table 2, the t10-Student copula seems to be the one which better approximate the empirical copula of the dataset. However, the difference with the Normal copula is very low. So the Gaussian copula could be appropriate. We remember that the use of the Gaussian copula permits us to construct algorithms to simulate scenarios from a multivariate distribution with different margins. The commonly used multivariate Normal is only a particular case where all the margins are Gaussians too.

The simulation algorithms showed in section 4 are applicated to simulate 1000 scenarios for the standardized log-returns of the TIM and the Olivetti equities, using the parameter estimations in Table 2. In all the cases, we have used standardized Gaussian margins, because our aim is only to compare the different copulas. In Figures 2, 3, 4, 5, 6, 7 and 8 we have plotted the results.

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Figure 2: 1000 Monte Carlo simulations of bivariate random variates (x1, x2) with Gaussian copula (R12=0.53248) and standard normal margins.

Figure 3: 1000 Monte Carlo simulations of bivariate random variates (x1, x2) with t20-Student copula (R12=0.53564) and standard normal margins.

Multivariate Normal CML

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t-20 Student Copula

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Figure 4: 1000 Monte Carlo simulations of bivariate random variates (x1, x2) with t10-Student copula (R12=0.54037) and standard normal margins.

Figure 5: 1000 Monte Carlo simulations of bivariate random variates (x1, x2) with t5-Student copula (R12=0.53953) and standard normal margins.

t10-Student Copula

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t5-Student Copula

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Figure 6: 1000 Monte Carlo simulations of bivariate random variates (x1, x2) with Clayton copula (α=1.12436) and standard normal margins.

Figure 7: 1000 Monte Carlo simulations of bivariate random variates (x1, x2) with Farlie-Gumbel-Morgenstern copula (α=1.55349) and standard normal margins.

Clayton Copula

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FGM Copula

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Figure 8: 1000 Monte Carlo simulations of bivariate random variates (x1, x2) with Frank copula (α=3.82211) and standard normal margins.

Comparing the plots in the above Figures with the one of the empirical distribution obtained from the historical data set in Figure 1, we can see some differences. These ones might be due to a wrong choice of the margins32. In fact, our aim was only to compare different types of copulas without assumptions about the analytical form of the marginal distributions.

Concluding remarks This paper is a review of the existing methodologies for calibrating, choosing and simulating different types of copulas.

The methods are applied to an historical dataset of Italian equity log-returns. We have seen how copula functions are a useful tool to construct efficient simulation algorithms. In fact, practical algorithms to generate Monte Carlo scenarios from a multivariate distribution with fixed copula and different margins are easily implemented to simulate financial asset returns. The traditional models use the multinormal distribution33 to simulate asset log- 32 Standardized Gaussians. 33 i.e. Gaussian copula and margins.

Frank Copula

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returns. Even using the Normal copula, we can choose different marginal distributions to construct more efficient algorithms. In fact the choice of the margins seems to have a more important impact than the choice of the copula on the results of the simulation. In this paper only the copula effects are taken into account. The choice of a more effective distributional form for the margins34 will be the aim of our further research.

34 e.g. α-Stable distributions or Generalized Hyperbolic distributions.

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