Copule slides

Modelling Dependence with Copulas
An Introduction
Giovanni Della Lunga
WORKSHOP IN QUANTITATIVE FINANCE
Bologna - May 3-4, 2018
Giovanni Della Lunga (WORKSHOP IN QUANTITATIVE FINANCE)Modelling Dependence with Copulas Bologna - May 3-4, 2018 1 / 84

Outline
1 Introduction
2 Joint Distributions
3 Copulas
Survival Copula
Density
4 Dependence Concepts
Linear Correlation Coeﬃcient
Concordance
Kendall’s tau and Spearman’s
rho
5 Copula Families
Gaussian copulas
t-copulas
Archimedean copulas
6 Tail Dependence
7 Bibliography

Introduction
Introduction

Introduction
Introduction
Many real-life situations can be modelled by a large number of
random variables which play a significant role, and such variates are
generally not independent.
Therefore, it is often of fundamental importance to be able to link the
marginal distributions of different variables in order to give a flexible
and accurate description of the joint law of the variables of interest.
Copulas were introduced in 1959 in the context of probabilistic metric
spaces and later exploited as a tool for understanding relationships
among multivariate outcomes.
A copula is a function that links univariate marginals to their joint
multivariate distribution in such a way that it captures the entire
dependence structure in the multivariate distribution.

Introduction
Introduction
The main advantage provided by a copula-approach in dependence
modelling is that the selection of an appropriate model for the
dependence between variables X and Y , represented by the copula,
can proceed independently from the choice of the marginal
distributions.
The seminal result in the history of copulas is due to Sklar that
introduced in 1959 the notion, and the name, of copula, and proved
the theorem that now bears his name (Sklar, 1959).
The latter states that any multivariate distribution can be expressed
as its copula function evaluated at its marginal distribution functions.
Moreover, any copula function when evaluated at any marginal
distributions is a multivariate distribution.

Introduction
Introduction
This presentation is so organized: in section 2 we recall some basic
concepts from multivariate distributions theory, after Section 3 in
which we deﬁne the concept of copula in full generality, we turn in
Section 4 to an overview of the most important notions of
dependence used in IRM. Section 5 introduces the most important
families of copulas, their properties both methodological as well as
with respect to simulation. Finally in Section 6 we discuss the
concept of tail dependence.
I would like to stress that this presentation only gives a ﬁrst
introduction, topics not included are statistical estimation of copulas
and the modelling of dependence, through copulas, in a dynamic
environment.

Introduction
The Problem (Embrechts, 2009)
A problem well known by all
those who deal with Risk
Management is the following:
”Here we are given a multi-
dimensional (i.e. several risk
factors) problem for which we
have marginal information
together with an idea of
dependence”.
Now the question is: When is
this problem well-posed?

Introduction
The Problem (Embrechts, 2009)
One concrete question could be this: given two marginal, one-period
risk factors X1 ed X2 with lognormal distribution functions (dfs)
F1 = LN(µ = 0, σ = 1) and F1 = LN(µ = 0, σ = 4). How can we
simulate from such a model if X1 and X2 have linear correlation
ρ = 0.5 say?
First of all, the correlation information say something (but what?)
about the bivariate df of the random vectors (X1, X2)T , i.e. about
F(x1, x2) = Prob[x1 ≤ X1, x2 ≤ X2];
Note however that, in the above, that information is not given; we
only know F1, F2 e ρ;
What else would one need?

Introduction
The easy Copula argument
First note that for random variables (rvs) Xi with continuous dfs Fi ,
variables Ui = Fi (Xi ) are uniformly distributed rvs on [0, 1];
Hence for some joint df F with marginal dfs F1 e F2 we have:
F(X1, X2) = P(X1 ≤ x1, X2 ≤ x2)
= P(F1(X1) ≤ F1(x1), F1(X2) ≤ F1(x2))
= P(U1 ≤ F1(x1), U2 ≤ F2(x2))
:= C(F1(x1), F2(x2))
(1)
The function C above is exactly a (careful here!) copula, a df on
I2 = [0, 1] × [0, 1] with standard uniform marginals, C is the df of the
random vector (U1, U2)T .

Introduction
The easy Copula argument
If we return to our lognormal example, we see no immediate reason
how the number ρ should determine the function C, it is not even
clear whether the problem has none, one or inﬁnitely many solutions;
In this case, it turns out that the problem posed has no solution.
The time has come to move on to some deﬁnition ...

Joint Distributions
Joint Distributions

Joint Distributions
Distribution Function
Lets take a random real value array X = (X1, . . . , Xd ) we can deﬁne a
multivariate distribution function F:
F(x) = Prob[X1 ≤ x1, . . . , Xd ≤ xd ]
In general a distribution function F deﬁned from R in I, must meet
the following conditions:
F(x) ≥ 0 ∀x
lim
xi →+∞
F(x1, . . . , xd ) = 1 ∀xi
lim
xi →−∞
F(x1, . . . , xd ) = 0 ∀xi

Joint Distributions
Distribution Function
Let X be a uniform random
variable [0, 1], the distribution
function is:
F(x) =



0 x < 0
x 0 ≤ x < 1
1 x = 1

Joint Distributions
Joint Cumulative Distribution Function
In particular the joint cumulative function of two random variables X
and Y is deﬁned as
FXY (x, y) = P(X ≤ x, Y ≤ y) (2)
The joint CDF satisﬁes the following properties:
1 FX (x) = FXY (x, ∞), for any x (marginal CDF of X);
2 FY (y) = FXY (∞, y), for any y (marginal CDF of Y );
3 FXY (∞, ∞) = 1;
4 FXY (−∞, y) = FXY (x, −∞) = 0;
5 P(x1 < X ≤ x2, y1 < Y ≤ y2) =
FXY (x2, y2) − FXY (x1, y2) − FXY (x2, y1) + FXY (x1, y1);
6 if X and Y are independent, then FXY (x, y) = FX (x)FY (y).

Joint Distributions
Joint Cumulative Distribution Function
In particular from property 5, putting x2 → +∞ and y2 → +∞ we have
P(x1 < X ≤ +∞, y1 < Y ≤ +∞) = FXY (+∞, +∞) − FXY (x1, +∞)
− FXY (+∞, y1) + FXY (x1, y1)
= 1 − FX (x) − FY (y) + FXY (x1, y1)
(3)
If we denote with ¯FXY (x, y) = P[X > x, Y > y], we ﬁnally obtain
¯FXY (x, y) = 1 − FX (x) − FY (y) + FXY (x, y) (4)

Copulas
Copulas

Copulas
Copulas
The concept of copula arises from the idea of breaking down a
multivariate distribution F into components allow one to easily model
and estimate the distribution of random vectors by estimating
marginals and dependency structure separately;
The importance of the copula functions derives entirely from a
noteworthy result known in the literature as Sklar’s theorem that
states that any multivariate joint distribution can be written in terms
of univariate marginal distribution functions and a copula which
describes the dependence structure between the variables;
Before discussing the theorem we are gong to describe some of the
fundamental properties of copula functions limiting (for simplicity) to
the bivariate case and recalling our deﬁnition of copula function as a
bi-variate distribution of two marginally uniform variables ...

Copulas
Copulas
According to our previous deﬁnition
C(x, y) = P(U1 ≤ x, U2 ≤ y) (5)
From the above equation we can write:
C(x, 0) = P(U1 ≤ x, U2 ≤ 0) = 0
C(0, y) = P(U1 ≤ 0, U2 ≤ y) = 0
C(x, 1) = P(U1 ≤ x, U2 ≤ 1) = P(U1 ≤ x) = x
C(1, y) = P(U1 ≤ 1, U2 ≤ y) = P(U2 ≤ y) = y
(6)
in particular from the last two relations of (6) we obtain
C(x, y) ≤ C(x, 1) = x C(x, y) ≤ C(1, y) = y ⇒ C(x, y) ≤ min(x, y) (7)

Copulas
Copulas
The volume of the square in the
picture is clearly equal to
V ([x1, x2] × [y1, y2]) =
(x2 − x1) · (y2 − y1) =
x2y2 − x2y1 − x1y2 + x1y1
(8)
Generaly speaking, for each
function H we can deﬁne a
generalized-volume or H-Volume
by:
VH(B) = H(x2, y2) − H(x2, y1)
− H(x1, y2) + H(x1, y1)
(9)

Copulas
Copulas
For the C-Volume to be considered as a probability, it must be always
positive deﬁnite
C(x2, y2) − C(x2, y1) − C(x1, y2) + C(x1, y1) ≥ 0 (10)
This must be true however the four values xi , yi are chosen, in
particular if we choose (x2, y2) = (x, y) e (x1, y1) = (1, 1) we have
C(x, y) − C(x, 1) − C(1, y) + C(1, 1) ≥ 0 (11)
Remember that C(x, 1) = x e C(1, y) = y, so we have
C(x, y) ≥ x + y − 1 and C(x, y) ≥ 0 then:
C(x, y) ≥ max(x + y − 1, 0) (12)

Copulas
Fréchet Limits
Putting together (12) and (7) we have
max(x + y − 1, 0) ≤ C(x, y) ≤ min(x, y) (13)
This is a special case of one of the most important results of
multivariate statistics theFréchet- Hoeffding Theorem.
Theorem (Fréchet-Hoeffding Bounds)
Suppose F1, . . . , Fd are marginal dfs and F any joint df with those given
marginals, then ∀x ∈ Rd ,
d
k=1
Fk(xk) + 1 − d
+
≤ F(x) ≤ min(F1(x1), . . . , Fd (xd )) (14)

Copulas
Fr´echet Limits
Figure: a) Minimum Copula Figure: b) Maximum Copula

Copulas
Sklar’s Theorem
The key to link copula functions and distribution functions is
represented by the Sklar Theorem:
Theorem (A. Sklar, 1959)
Suppose X1, . . . , Xd are rvs with continuous dfs F1, . . . , Fd and joint df F,
then there exists a unique copula C (a df on [0, 1]d with uniform
marginals) such that for all x ∈ Rd :
F(x1, . . . , xd ) = C(F1(x1), . . . , Fd (xd )) (15)
Conversely given any dfs F1, . . . , Fd and copula C, F deﬁned through (15)
is a d-variate df with marginal dfs F1, . . . , Fd .

Copulas
Sklar’s Theorem
The theorem guarantees that for continuous random variables, the
univariate margins and the multivariate dependence structure can be
univocally separated and that the copula fully describes the
dependence structure;
The theorem can be reversed, so that the copula can be described in
terms of a joint distribution function and quantile functions of the
marginal ones.
Corollary
Let be F, F1, . . . , Fd and C the same as in Sklar Theorem and
F−1
1 , . . . , F−1
d the quantile functions of F1, . . . , Fd then we can write:
C(u) = F(F−1
1 (u1), . . . , F−1
d (ud )) (16)

Copulas
Coding Interlude - How a Copula Works ...

Copulas
Independence
Also the concept of
independence between random
variables can be represented in
terms of copulas;
In terms of copulas, if the
random variables are
independent, the functional that
describes the link between the
marginal distribution functions
and the joint one is the product
copula, conversely if some
variables are associated with the
product copula then they are
independent

Copulas Survival Copula
Survival Copula
The survival copula associated with the copula C is
¯C(u, v) = u + v − 1 + C(1 − u, 1 − v) (17)
It is easy to verify that ¯C ha the copula properties. Once computed in
(1 − u, 1 − v) is equivalent to the complementary distribution
function of a bivariate uniform distribution, since
¯C(1 − u, 1 − v) = 1 − u + 1 − v − 1 + C(u, v)
= 1 − P(U1 ≤ u) + 1 − P(U2 ≤ v) − 1
+ P(U1 ≤ u, U2 ≤ v)
= 1 − P(U1 ≤ u) − P(U2 ≤ v) + P(U1 ≤ u, U2 ≤ v)
= P(U1 > u, U2 > v)
(18)

Copulas Density
Density
If C is absolutely continuous then it can be written in the form
C(u) =
[0,u]d
c(s) ds (19)
where c is a suitable function called density of C.
In particular, for almost all u ∈ Id one has
c(u) =
∂d C(u)
∂u1 . . . ∂ud
(20)
As stressed by many authors, this equation is far from obvious. In
fact, there are some facts that are implicitly used: ﬁrst, the mixed
partial derivatives of order d of C exist and are equal almost
everywhere on Id ; second each mixed partial derivative is actually
almost everywhere equal to the density c.

Copulas Density
Density
We will use this result to formally deﬁne a measure induced on Id by
C.
dC(u) = c(u) du (21)
In particular for a bivariate copula we have
dC(u, v) =
∂2C(u, v)
∂u∂v
dudv (22)

Copulas Density
Copula and Measure: Example
Assume that C is absolutely continuous. The following identity is true
[0,1]2
uvdC(u, v) =
[0,1]2
C(u, v) dudv (23)
If C is absolutely continuous then we can write
dC(u, v) =
∂2C(u, v)
∂u∂v
dudv (24)
Sostitution of this deﬁnition in the left hand member and evaluating
the inner integrals by parts give us...

Copulas Density
1
0
1
0
uvdC(u, v) =
1
0
1
0
uv
∂2
C(u, v)
∂u∂v
dudv =
1
0
du u
1
0
dv v
∂2
C(u, v)
∂u∂v
=
1
0
du u v
∂C(u, v)
∂u
v=1
v=0
−
1
0
∂C(u, v)
∂u
dv =
1
0
du u 1 −
1
0
∂C(u, v)
∂u
dv
=
1
0
du u −
1
0
dv
1
0
du u
∂C(u, v)
∂u
=
1
0
du u −
1
0
dv uC(u, v)
u=1
u=0
+
1
0
C(u, v) du
=
1
0
du u −
1
0
dv v +
1
0
1
0
C(u, v) dudv =
1
0
1
0
C(u, v) dudv
(25)

Copulas Density
Exercise Compute the following expression
1
0
1
0
C(u, v)
∂2C(u, v)
∂u∂v
du dv (26)

Copulas Density
Answer Evaluate the inner integral by parts
1
0
C(u, v)
∂2
∂u∂v
C(u, v) du
= C(u, v)
∂
∂v
C(u, v)
u=1
u=0
−
1
0
∂
∂u
C(u, v)
∂
∂v
C(u, v) du
= v −
1
0
∂
∂u
C(u, v)
∂
∂v
C(u, v) du
(27)
performing the second integral we have
1
0
1
0
C(u, v)
∂2
∂u∂v
C(u, v) dudv
=
1
2
−
1
0
1
0
∂
∂u
C(u, v)
∂
∂v
C(u, v) dudv.
(28)

Dependence Concepts
Dependence Concepts

Dependence Concepts
Dependence Measures
Copulas provide a natural way to study and measure dependence
between random variables.
Copula properties are invariant under strictly increasing
transformations of the underlying random variables.
Linear correlation (or Pearson’s correlation) is most frequently used in
practice as a measure of dependence. However, since linear
correlation is not a copula-based measure of dependence, it can often
be quite misleading and should not be taken as the canonical
dependence measure. In the next slides we recall the basic properties
of linear correlation, and then continue with some copula based
measures of dependence.

Dependence Concepts Linear Correlation Coefficient
Definition. Let (X, Y )T be a vector of random variables with nonzero
finite variances. The linear correlation coefficient for (X, Y )T is
ρ(X, Y ) =
Cov(X, Y )
Var(X) Var(Y )
(29)
where Cov(X, Y ) = E(XY ) − E(X)E(Y ) is the covariance of (X, Y )T ,
and Var(X) and Var(Y ) are the variances of X and Y .

Linear correlation is a popular but also often misunderstood measure
of dependence.
The popularity of linear correlation stems from the ease with which it
can be calculated and it is a natural scalar measure of dependence in
elliptical distributions (with well known members such as the
multivariate normal and the multivariate t- distribution).
However most random variables are not jointly elliptically distributed,
and using linear correlation as a measure of dependence in such
situations might prove very misleading.
Even for jointly elliptically distributed random variables there are
situations where using linear correlation does not make sense. We
might choose to model some scenario using heavy-tailed distributions
such as t2-distributions. In such cases the linear correlation coefficient
is not even defined because of infinite second moments.

Property. ρ(X, Y ) is bounded:
ρl ≤ ρ ≤ ρu
where the bounds ρl and ρu are deﬁned as
ρl =
D
C−
(F1(x), F2(y)) − F1(x)F2(y) dx dy
Dom(F1)
(x − E(x))2dF1(x) Dom(F2)
(y − E(y))2dF2(y)
ρu =
D
C+
(F1(x), F2(y)) − F1(x)F2(y) dx dy
Dom(F1)
(x − E(x))2dF1(x) Dom(F2)
(y − E(y))2dF2(y)
(30)
and are attained respectively when X and Y are countermonotonic and
comonotonic.

Proof. The bound for ρ(X, Y ) can be obtained directly from Hoeﬀding’s
expression for covariance together with the Fr´echet inequality.
Example. Let X ∼ LN(0, σ2
1) and Y ∼ LN(0, σ2
2). Then
ρmin = ρ(eσ1Z , e−σ2Z ) and ρmax = ρ(eσ1Z , eσ2Z ), where Z ∼ N(0, 1). ρmin
and ρmax can be computed yielding:
ρl =
exp(−σ1σ2) − 1
(exp(σ2
1) − 1) (exp(σ2
2) − 1)
≤ 0
ρu =
exp(σ1σ2) − 1
(exp(σ2
1) − 1) (exp(σ2
2) − 1)
≥ 0
(31)

Theoretical Interlude - How compute ρ for log-Normal
distribution.

Note that
lim
σ→∞
ρmin = lim
σ→∞
ρmax = 0

Dependence Concepts Concordance
Concordance
Concordance concepts, loosely speaking, aim at capturing the fact
that the probability of having ”large” (or ”small”) values of both X
and Y is high, while the probability of having ”large” values of X
together with ”small” values of ”Y” - or viceversa - is low.
Let (x, y)T and (˜x, ˜y)T be two observations from a vector (X, Y )T of
continuous random variables.
Then (x, y)T and (˜x, ˜y)T are said to be concordant if
(x − ˜x)(y − ˜y) > 0, and discordant if (x − ˜x)(y − ˜y) < 0.
The following theorem can be found in Nelsen (1999) p. 127. Many
of the results in this section are direct consequences of this theorem.

Concordance
Theorem
Let (X, Y )T and ( ˜X, ˜Y )T be independent vectors of continuous random
variables with joint distribution functions H and ˜H, respectively, with
common margins F (of X and ˜X) and G (of Y and ˜Y ). Let C and ˜C
denote the copulas of (X, Y )T and ( ˜X, ˜Y )T respectively,so that
H(x, y) = C(F(x), G(y)) and ˜H(x, y) = ˜C(F(x), G(y)). Let Q denote
the diﬀerence between the probability of concordance and discordance of
(X, Y )T and ( ˜X, ˜Y )T , i.e. let
Q = P[(X − ˜X)(Y − ˜Y ) > 0] − P[(X − ˜X)(Y − ˜Y ) < 0] (32)
Then
Q = Q(C, ˜C) = 4
[0,1]2
˜C(u, v)dC(u, v) − 1 (33)

Concordance
Proof. Since the random variables are all continuous,
P[(X − ˜X)(Y − ˜Y ) < 0] = 1−P[(X − ˜X)(Y − ˜Y ) > 0] ⇒ Q = 2P[(X − ˜X)(Y − ˜Y ) > 0]−1
But
P[(X − ˜X)(Y − ˜Y ) > 0] = P[X > ˜X, Y > ˜Y ] + P[X < ˜X, Y < ˜Y ]
and these probabilities can be evaluated by integrating over the
distribution of one of the vectors (X, Y )T or (˜X, ˜Y )T . Hence
P[X > ˜X, Y > ˜Y ] = P[ ˜X < X, ˜Y < Y ]
=
R2
P[ ˜X < x, ˜Y < y] dC[F(x), G(y)]
=
R2
˜C[F(x), G(y)] dC[F(x), G(y)]
(34)

Concordance
Employing the probability-integral transform u = F(x) and v = G(y) then
yields
P[X > ˜X, Y > ˜Y ] = P[ ˜X < X, ˜Y < Y ] =
[0,1]2
˜C(u, v) dC(u, v) (35)
Similarly,
P[X < ˜X, Y < ˜Y ] =
R2
P[ ˜X > x, ˜Y > y] dC[F(x), G(y)]
=
R2
1 − F(x) − G(y) + ˜C[F(x), G(y)] dC[F(x), G(y)]
=
[0,1]2
1 − u − v + ˜C(u, v) dC(u, v)
(36)

Concordance
But since C is the joint distribution function of a vector (U, V )T
of U(0, 1) random
variables, E(U) = E(V ) = 1/2, and hence
P[X < ˜X, Y < ˜Y ] = 1 −
1
2
−
1
2
+
[0,1]2
˜C(u, v) dC(u, v) =
[0,1]2
˜C(u, v) dC(u, v) (37)
Thus
P[(X − ˜X)(Y − ˜Y ) > 0] = 2
[0,1]2
˜C(u, v) dC(u, v) (38)
and the conclusion follows
Q = 2P[(X − ˜X)(Y − ˜Y ) > 0] − 1 = 4
[0,1]2
˜C(u, v) dC(u, v) − 1 (39)

Dependence Concepts Kendall’s tau and Spearman’s rho
Kendall’s tau and Spearman’s rho
Deﬁnition. Kendall’s tau for the random vector (X, Y )T is deﬁned as
τ(X, Y ) = P[(X − ˜X)(Y − ˜Y ) > 0] − P[(X − ˜X)(Y − ˜Y ) < 0] (40)
where ( ˜X, ˜Y )T is an independent copy of (X, Y )T . Hence Kendall’s tau
for (X, Y )T is simply the probability of concordance minus the probability
of discordance and since the copula of ( ˜X, ˜Y )T is the same of (X, Y )T is
also simply equal to Q(C, C):

Theorem. Let (X, Y )T be a vector of continuous random variables with
copula C. Then Kendall’s tau for (X, Y )T is given by
τ(X, Y ) = Q(C, C) = 4
[0,1]2
C(u, v) dC(u, v) − 1 (41)
Note that the integral above is the expected value of the random variable
C(U, V ), where U, V ∼ U(0, 1) with joint distribution function C, i.e.
τ = 4E(C(U, V )) − 1.

Deﬁnition. Spearman’s rho for the random vector (X, Y )T
is deﬁned as
ρS (X, Y ) = 3(P[(X − ˜X)(Y − Y ) > 0] − P[(X − ˜X)(Y − Y ) < 0]) (42)
where (X, Y )T
, ( ˜X, ˜Y )T
and (X , Y )T
are independent copies.
Theorem. Let (X, Y )T
be a vector of continuous random variables with copula C.
Then Spearman’s rho for (X, Y )T
is given by (note that ˜X and Y are
independent so their copula is the product copula)
ρS (X, Y ) = 3Q(C, Π) = 12
[0,1]2
uv dC(u, v) − 3 = 12
[0,1]2
C(u, v) du dv − 3
=
E(UV ) − 1/4
1/12
=
Cov(U, V )
Var(U) Var(V )
= ρ[F(X), G(Y )]
(43)

Copula Families
Copula Families

Copula Families Gaussian copulas
Elliptical Copulas
The class of elliptical distributions provides a rich source of
multivariate distributions which share many of the tractable properties
of the multivariate normal distribution and enables modelling of
multivariate extremes and other forms of nonnormal dependences.
Elliptical copulas are simply the copulas of elliptical distributions.
Simulation from elliptical distributions is easy, and as a consequence
of Sklar’s Theorem so is simulation from elliptical copulas.
Furthermore, we will show that rank correlation and tail dependence
coeﬃcients can be easily calculated

Gaussian Copula
The copula of the n-variate normal distribution with linear correlation
matrix ρ is
CGa
ρ (u1, . . . , u2) = Φd
ρ (φ−1
(u1), . . . , φ−1
(ud ))
where Φd
ρ denotes the joint distribution function of the n-variate
standard normal distribution function with linear correlation matrix ρ,
and φ−1 denotes the inverse of the distribution function of the
univariate standard normal distribution.
Copulas of the above form are called Gaussian copulas.

Gaussian Copula
We now address the question of random variate generation from the
Gaussian copula CGa
ρ .
For our purpose, it is suﬃcient to consider only strictly positive
deﬁnite matrices Σ. Write Σ = AAT for some n × n matrix A, and if
Z1, ..., Zn ∼ N(0, 1) are independent, then µ + AZ ∼ Nn(µ, Σ).
One natural choice of A is the Cholesky decomposition of Σ. The
Cholesky decomposition of Σ is the unique lower-triangular matrix L
with LLT = Σ. Furthermore Cholesky decomposition routines are
implemented in most mathematical software.
This provides an easy algorithm for random variate generation from
the Gaussian n-copula CGa
ρ .

Gaussian Copula: Simulation Algorithm
Find the Cholesky decomposition A of Σ.
Simulate n independent random variates z1, ..., zn from N(0, 1).
Set x = Az.
Set ui = Φ(xi ), i = 1, ..., n.
(u1, ..., un)T ∼ CGa
ρ
As usual Φ denotes the univariate standard normal distribution function.

Gaussian Copula with Gaussian Marginals (ρ = 0.2)

Gaussian Copula with tν Marginals (ρ = 0.2)

Gaussian Copula with Gaussian Marginals (ρ = 0.9)

Gaussian Copula with tν Marginals (ρ = 0.9)

Copula Families t-copulas
t copula
Similarly to the case of a normal distribution we can consider the distribution tν ;
the corresponding copula with correlation ρ is:
Ct
ν,ρ(u1, . . . , ud ) = Θd
ν,ρ(t−1
ν (u1), . . . , t−1
ν (ud )) (44)
where Θd
ν,ρ is the t multivariate distribution with dimension d, ν degree of
freedom and linear correlation ρ while t−1
ν is the inverse univariate standard
t-distribution tν .
Also for this functional you have a simple simulation algorithm based on the
following result: If X has the stochastic representation
X =
√
ν
√
Z
Y with Y ∼ Nd (0, ρ), Z ∼ χ2
ν
then X has an n − variate tν -distribution with mean µ (for ν > 1) and covariance
matrix νΣ/(ν − 2) (for ν > 2). If ν ≤ 2 then Cov(X) is not deﬁned. In this case
we just interpret Σ as being the shape parameter of the distribution of X.

t copula: Simulation Algorithm
Find the Cholesky decomposition A of Σ.
Simulate n independent random variates z1, ..., zn from N(0, 1).
Simulate a random variate s from χ2
ν independent of z1, ..., zn.
Set x = Az
Set y = n
s x
Set ui = Tν(yi ) for i = 1, . . . , n
(u1, ..., un)T ∼ Ct
ν,Σ.

Coding Interlude - Elliptical Copulas ...

Copula Families Archimedean copulas
Archimedean copulas
In this section we discuss an important class of copulas called
Archimedean copulas.
They have proved to be useful in several applications since they are
capable of capturing wide ranges of dependence structures.
Furthermore, in contrast to elliptical copulas, all commonly
encountered Archimedean copulas have closed form expressions.
We divide the discussion of Archimedean copulas in two subsections:
the ﬁrst introduces them and their main properties while the second
presents diﬀerent one-parameter families of Archimedean copulas.

Archimedean copulas
Archimedean copulas may be constructed using a function
φ : I → [0, ∞], continuous, decreasing, convex and such that
φ(1) = 0.
Such a function φ is called a generator. It is called a strict generator
whenever φ(0) = +∞.
The pseudo-inverse of φ is deﬁned as follows:

Archimedean copulas
The pseudo-inverse is such that, by composition with the generator, it
gives the identity, and it coincides with the usual inverse if φ is a strict
generator.
Deﬁnition Given a generator and its pseudo-inverse, an Archimedean
2-copula takes the form
C(u, v) = φ[−1]
φ(u) + φ(v) (45)
If the generator is strict, the copula is said to be a strict Archimedean
2-copula.

Archimedean copulas
Among Archimedean copulas, we are going to consider in particular
the one-parameter ones, which are constructed using a generator
φα(t), indexed by a real parameter α;
By choosing the generator, one obtains a subclass or family of
Archimedean copulas;
The following table describes some well-known families and their
generator (for a more exhaustive list see Nelsen , 1999)

Archimedean copulas

Archimedean copulas: Gumbel d-copula
The Gumbel family has been introduced by Gumbel (1960). Since it has
been discussed in Hougaard (1986), it is also known as the
Gumbel–Hougaard family. The standard expression for members of this
family of d-copulas is
The case α = 1 gives the product copula as a special case, and the limit
for α → +∞ is the comonotonicity copula. It follows that the Gumbel
family can represent independence and positive dependence only. The
generator is given by
φα(u) = − log u
α
, α ≥ 1 (46)

Archimedean copulas: Clayton d-copula
The Clayton family was ﬁrst proposed by Clayton (1978), and studied by
Oakes (1982). The standard expression for members of this family of
d-copulas is
The limiting case α = 0 corresponds to the independence copula. The
generator has the form
φα(u) = u−α
− 1, α > 0 (47)

Archimedean copulas: Frank d-copula
Copulas of this family have been introduced by Frank (1979), and have
the expression:
It reduces to the product copula if α = 0. For the case d = 2, the
parameter α can be extended also to the case α < 0. The generator is
given by

Bivariate sample from the Gumbel copula

Bivariate sample from the Clayton copula

Bivariate sample from the Frank copula

Coding Interlude - R Gumbel Package ...

Tail Dependence
Tail Dependence

Tail Dependence
Tail Dependence
The concept of tail dependence relates to the amount of dependence
in the upper-right- quadrant tail or lower-left- quadrant tail of a
bivariate distribution.
It is a concept that is relevant for the study of dependence between
extreme values.
It turns out that tail dependence between two continuous random
variables X and Y is a copula property and hence the amount of tail
dependence is invariant under strictly increasing transformations of X
and Y .

Tail Dependence
Tail Dependence

Tail Dependence
Tail Dependence
For copulas without a simple closed form an alternative formula for
λU is more useful.
Consider a pair of U(0, 1) random variables (U, V ) with copula C.
First note that
P(V ≤ v | U = u) =
∂C(u, v)
∂u
(48)
and
P(V > v | U = u) = 1 −
∂C(u, v)
∂u
(49)
and similarly when conditioning on V

Tail Dependence
Tail Dependence
Remember that, by deﬁnition
d ¯C(u, u)
du
= lim
∆u→0
¯C(u + ∆u, u + ∆u) − ¯C(u, u)
∆u
let’s assume ∆u = 1 − u
d ¯C(u, u)
du
= lim
u→1
¯C(1, 1) − ¯C(u, u)
1 − u
= − lim
u→1
¯C(u, u)
1 − u
since ¯C(1, 1) = 0

Tail Dependence
Tail Dependence
We can write
λU = lim
u→1
¯C(u, u)
1 − u
= − lim
u→1
d ¯C(u, u)
du
= lim
u→1
−2 +
∂C(s, t)
∂s s=t=u
+
∂C(s, t)
∂t s=t=u
= lim
u→1
P(V > u | U = u) + P(U > u | V = u)
(50)
Furthermore if C is an exchangeable copula, i.e. C(u, v) = C(v, u),
then the expression for λU simpliﬁes to
λU = 2 lim
u→1
P(V > u | U = u) (51)

Tail Dependence
Tail Dependence: Gaussian Copula
Let (X, Y )T
have the bivariate standard normal distribution function with linear
correlation coeﬃcient ρ. That is (X, Y )T
∼ C(Φ(x), Φ(y)), where C is a member of the
gaussian family. Since copulas in this family are exchangeable and because Φ is a
distribution function with inﬁnite right endpoint,
lim
u→1
P V > u | U = u = lim
x→∞
P Φ−1
(V ) > x | Φ−1
(U) = x = lim
x→∞
P X > x | Y = x
(52)
Using the well known fact that (Y | X = x) ∼ N(ρx, 1 − ρ2
) we obtain
λU = 2 lim
x→∞
¯Φ (x − ρx)/ 1 − ρ2 = 2 lim
x→∞
¯Φ x 1 − ρ/ 1 + ρ (53)
from which it follows that λU = 0 for ρ < 1, hence the gaussian copula does not have
upper tail dependence.

Tail Dependence
Tail Dependence: t-copula
If (X1, X2)T has a standard bivariate t-distribution with ν degrees of
freedom and linear correlation matrix R, then (X2|X1 = x) is t-distributed
with ν + 1 degrees of freedom and
E(X2|X1 = x) = R12x, Var(X2|X1 = x) =
ν + x2
ν + 1
1 − R2
12
This can be used to show that the t-copula has upper (and because of
radial symmetry equal lower) tail dependence

Tail Dependence
Tail Dependence: t-copula
Let’s compute the λU factor:
λU = 2 lim
x→∞
P X2 > x|X1 = x
= 2 lim
x→∞
¯tν+1
ν + 1
ν + x2
1/2 x − R12x
1 − R2
12
= 2 lim
x→∞
¯tν+1
ν + 1
ν/x2 + 1
1/2
√
1 − R12
√
1 + R12
= 2¯tν+1
√
ν + 1
√
1 − R12
√
1 + R12
(54)
From this it is also seen that the coefficient of upper tail dependence is increasing
in R12 and decreasing in ν. Furthermore, the coefficient of upper (lower) tail
dependence tends to zero as the number of degrees of freedom tends to infinity for
R12 < 1.

Bibliography
Bibliography

Bibliography
Bibliography
U. Cherubini et al. ”Copula Methods in Finance”, Wiley Finance, 2013
Embrechts ”Copulas a Personal View”
Embrechts et al. ”Modelling Dependence with Copulas and Application to Risk
Management”, Department of Mathematik, ETHZ, Zurich, working paper
Embrechts et al. ”Correlation and Dependency in Risk Management: Properties
and Pitfalls”, Department of Mathematik, ETHZ, Zurich, working paper
Lindskog ”Linear Correlation Estimation”, RiskLab ETH, working paper
Nelsen ”An Introduction to Copulas”, Lecture Notes in Statistics, Springer-Verlag,
1999
Scarsini ”On Measures of Concordance”, Stochastica, 8, 201-208

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