2.1 Copulas
In this section we summarize the basic results without proof that are
necessary to understand the concept of copulas. Then, we present the most
important properties of copulas that are needed for applications in finance.
In doing so, we will follow the notation used in Nelsen (1999).
2.1.1 Definition
DEFINITION 2.1
A
2-dimensional copula is a function
![$ C: \, [0,1]^2 \to [0,1]$](xfghtmlimg460.gif)
with the following properties:
- For every
 |
(2.1) |
- For every
 |
(2.2) |
- For every
with
and
:
 |
(2.3) |
A function that fulfills property 1 is also said to be grounded.
Property 3 is the two-dimensional analogue of a nondecreasing
one-dimensional function. A function with this feature is therefore called
2-increasing.
The usage of the name ''copula'' for the function
is explained by the
following theorem.
2.1.2 Sklar's Theorem
The distribution function of a random variable
is a
function
that assigns all
a probability
. In addition, the joint distribution
function of two random variables
is a function
that
assigns all
a probability
.
THEOREM 2.1 (Sklar's theorem)
Let

be a joint distribution function with margins

and

. Then there exists a copula

with
 |
(2.4) |
for every

. If

and

are
continuous, then

is unique. Otherwise,

is uniquely determined
on Range

Range

. On the other hand, if

is a copula and

and

are distribution functions, then
the function

defined by (
2.4) is a joint distribution
function with margins

and

.
It is shown in Nelsen (1999) that
has margins
and
that
are given by
and
, respectively. Furthermore,
and
themselves are distribution functions.
With Sklar's Theorem, the use of the name ``copula''
becomes obvious.
It was chosen by Sklar (1996) to describe ``a function that
links a multidimensional distribution to its one-dimensional
margins'' and appeared in mathematical literature for the first
time in Sklar (1959).
2.1.3 Examples of Copulas
The
structure of independence is especially important for
applications.
DEFINITION 2.2
Two random variables

and

are
independent
if and only if the product of their distribution functions

and

equals their joint distribution function

,
 |
(2.5) |
Thus, we obtain the independence copula
by
which becomes obvious from the following theorem:
THEOREM 2.2
Let

and

be random variables with continuous
distribution functions

and

and joint distribution
function

.
Then

and

are independent if and only if

.
From Sklar's Theorem we know that there exists a
unique copula
with
 |
(2.6) |
Independence can be seen using Equation (2.4) for the joint
distribution function
and the definition of
,
 |
(2.7) |
2.1.3.0.2 Gaussian Copula
The second important copula that we want to investigate is the
Gaussian or normal copula,
 |
(2.8) |
see Embrechts, McNeil and Straumann (1999). In (2.8),
denotes the bivariate normal density
function with correlation
for
. The functions
,
in (2.8) refer to the corresponding
one-dimensional, cumulated normal density functions of the margins.
In the case of vanishing correlation,
, the Gaussian copula becomes
Result (2.9) is a direct consequence
of Theorem 2.2.
As
, one can replace
in
(2.8) by
. If one
considers
in a probabilistic sense, i.e.
and
being values of two random variables
and
, one obtains
from (2.8)
 |
(2.10) |
In other words:
is the
binormal cumulated probability function.
2.1.3.0.3 Gumbel-Hougaard Copula
Next, we consider the Gumbel-Hougaard family of
copulas, see Hutchinson (1990). A discussion in Nelsen (1999) shows
that
is suited to describe bivariate extreme value distributions.
It is given by the function
![$\displaystyle C_{\theta}(u, v) \stackrel{\mathrm{def}}{=}\exp \left\{ - \left[ (-\ln u)^{\theta} + (-\ln v)^{\theta} \right]^{1 / \theta} \right\} \; .$](xfghtmlimg508.gif) |
(2.11) |
The parameter
may take all values in the interval
.
For
, expression (2.11)
reduces to the product copula, i.e.
.
For
one finds for the Gumbel-Hougaard copula
It can be shown that
is also a copula. Furthermore, for any given
copula
one has
, and
is called the
Fréchet-Hoeffding upper bound.
The
two-dimensional function
defines a
copula with
for any other copula
.
is
called the Fréchet-Hoeffding lower bound.
2.1.4 Further Important Properties of Copulas
In this section we focus on the properties of copulas.
The theorem we will present next establishes the continuity of copulas
via a Lipschitz condition on
:
THEOREM 2.3
Let

be a copula. Then for every
![$ u_1, u_2, v_1, v_2 \in [0,1]$](xfghtmlimg520.gif)
:
 |
(2.12) |
From (2.12) it follows that every copula
is uniformly continuous on its domain.
A further important property of copulas concerns the partial derivatives
of a copula with respect to its variables:
THEOREM 2.4
Let

be a copula. For every
![$ u \in [0,1]$](xfghtmlimg461.gif)
, the partial
derivative

exists for almost
every
![$ v \in [0,1]$](xfghtmlimg523.gif)
. For such

and

one has
 |
(2.13) |
The analogous statement is true for the partial derivative

.
In addition, the functions

and

are
defined and nondecreasing almost everywhere on [0,1].
To give an example of this theorem, we consider the partial derivative
of the Gumbel-Hougaard copula (2.11) with respect
to
,
Note that for
and
for all
where
,
is a
strictly increasing function of
. Therefore the inverse function
is well defined. However, as one might guess from
(2.14),
can not be calculated
analytically so that some kind of numerical algorithm has to be used
for this task. As
is symmetric in
and
, the partial
derivative of
with respect to
shows an identical
behaviour for the same set of parameters.
We will end this section with a statement on the behaviour of copulas
under strictly monotone transformations of random variables.
THEOREM 2.5
Let

and

be random variables with continuous
distribution functions and with copula

. If

and

are strictly increasing functions on Range

and Range

, then

.
In other words:

is invariant under strictly increasing
transformations of

and

.