2.3 The semi-analytic approach: one factor Gaussian
Copula model
2.3.1 Copula functions: basic properties
Copula functions are useful tools for modelling dependency
between random variables, for they allow to separate the univariate margins and
the dependence structure from the multivariate distribution.
Theorem 1. Sklar's Theorem
Let F be a joint distribution function with margins F1, .., Fd.
There exists a copula function C such that for all x1, .., xd in [-8,
+8],
F(x1,..,xd) = C(F1(x1),..,Fd(xd))
Conversely, if C is a copula function and F1, .., Fd are the
margins of respectively X1, .., Xd, then the multivariate function F
of the vector (X1, ..Xd) is such that, for all x1, .., xd in [-8,
+8],
F (x1, .., xd) = C(F1(x1), .., Fd(xd))
If the margins are continuous, then the copula function C is
unique.
We shall need another key result in order to ensure copula
functions are flexible enough to model joint loss distributions:
Proposition 1. Invariance
Let C denote the copula function of continuous random vector
(X1, .., Xd). Let
f1, .., fd be strictly increasing functions
defined respectively on the support of X1, .., Xd.
Then C is also
the copula function of the continuous random vector (f(X1), .., f(Xd)).
We then recall the cumulative distribution function Ö of
a standard gaussian variable and that of a multivariate standard centered
gaussian vector with correlation matrix R:
|
x 1
Ö(x) =
f8v2ð
|
e-t2/2dt
|
x1xd11T 0-1y · dy1..dyd
ÖV xd) "
f8- f e2y
Definition 1. Gaussian Copula
Let (X1, .., Xd) be a gaussian vector with correlation matrix R,
zero mean and unit variance. We can then express its copula function
CR as follows:
CR(u1, .., ud) = Ö`V/)
(Ö-1(u1),..,Ö-1(ud))
Given the invariance property of copula functions seen in
proposition (1), CR is also the copula function of any gaussian vector with
correlation matrix R.
2.3.2 The one factor gaussian copula model
Let (ô1, .., ôN) define the random vector of
default times among the N obligors of
our reference portfolio. Given
equation (2.1) and under deterministic assumptions
for recovery rates,
determining the joint distribution of (ô1, .., ôN) is equivalent
to
determining the joint loss distribution L(t) for all t = T.
We further assume that each default time random variable
ôj, j = 1..N, follows an exponential law of parameter ëj. In other
words, the cumulative distribution function Qj of ôj can be expressed
as:
?t ? [0, T], P(ôj = t) := Q(t) = 1 - e_ëjt
We now wish to model the dependency between those default time
random variables. The current market standard for doing so is to use the
gaussian copula function CR where its correlation matrix R is defined as
follows:
?
? ? ? ? ?
R=
?
?????
1 p ... p
p 1 ..
.. ..
... ...
. .. p
p ... p 1
Applying Sklar's reciprocal theorem, we can then exhibit the
resulting cumulative distribution function Q of the random vector (ô1,
.., ôN):
P(ô1 = t1,..,ôN = tN) := Q(t1,..,tN) = CR
(Q1(t1),..,QN(tN))
A convenient way to simulate the random vector of default
times (ô1, .., ôN) related together by a gaussian copula is to use
an auxiliary random vector (X1, .., XN) modelled upon a single factor approach.
We assume that all Xj, j = 1..N, depend respectively on a common standard
gaussian factor Z and on an idiosyncratic standard gaussian factor Zj, where
all Zj are mutually independent and independent from Z. Conditionnally on the
common factor Z, all Xj, j = 1, .., N are therefore independent.
?j ? [1,..,N],Xj := vpZ + /1 - pZj
Proposition 2. The random vector (X1, .., XN) is a gaussian
vector with correlation matrix equal to R.
Proof.
v/
?
????????
=
...
0 vp
...
...
. ..
0
....
. .. ..
Z
Z1
...
·
?
???????
...
ZN
?
???????
?
????????
(X1, .., XN) = (vpZ + v/ 1 - pZ1, .., vpZ + 1 - pZN) vp -v1 - p 0
... 0
...
....
. .. .. -v1 - p
0 vp
0 . . .
We have expressed (X1, .., XN) as an affine transformation of
the gaussian vector (Z, Z1, .., ZN). Hence, (X1, .., XN) is a gaussian vector
itself. The general term of its correlation matrix (pij,1 = i, j = N), is given
by:
Cov (vpZ + v1 - pZi,vpZ + v1 - pZj) pij = V ar(Xi)V
ar(Xj)
= äij(1 - p) + p
Hence, the correlation matrix of (X1, .., XN) is also R.
Applying invariance proposition (1) to the random vector of
default times (ô1,.., ôN) Law = (Q-1
1 (Ö(X1)), .., Q-1
N (Ö(XN)), we conclude that both vectors share the same
gaussian copula function.
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