Download Lecture 5 – Copula functions

Copula functions
Advanced Methods of Risk Management
Umberto Cherubini
Copula functions
• Copula functions are based on the principle of
integral probability transformation.
• Given a random variable X with probability
distribution FX(X). Then u = FX(X) is uniformly
distributed in [0,1]. Likewise, we have v = FY(Y)
uniformly distributed.
• The joint distribution of X and Y can be written
H(X,Y) = H(FX –1(u), FY –1(v)) = C(u,v)
• Which properties must the function C(u,v) have
in order to represent the joint function H(X,Y) .
Copula function
Mathematics
• A copula function z = C(u,v) is defined as
1. z, u and v in the unit interval
2. C(0,v) = C(u,0) = 0, C(1,v) = v and C(u,1) = u
3. For every u1 > u2 and v1 > v2 we have
VC(u,v) 
C(u1,v1) – C (u1,v2) – C (u2,v1) + C(u2,v2)  0
• VC(u,v) is called the volume of copula C
Copula functions:
Statistics
• Sklar theorem: each joint distribution
H(X,Y) can be written as a copula function
C(FX,FY) taking the marginal distributions
as arguments, and vice versa, every
copula function taking univariate
distributions as arguments yields a joint
distribution.
Copula function and
dependence structure
• Copula functions are linked to non-parametric dependence
statistics, as in example Kendall’s  or Spearman’s S
• Notice that differently from non-parametric estimators, the linear
correlation  depends on the marginal distributions and may not
cover the whole range from – 1 to + 1, making the assessment of
the relative degree of dependence involved.

 
  H x, y   F x F  y dxdy
X
  
1 1
 S  12  C u, v dudv  3
0 0
1 1
  4  C u, v dC u, v   1
0 0
Y
Dualities among copulas
• Consider a copula corresponding to the probability of the
event A and B, Pr(A,B) = C(Ha,Hb). Define the marginal
probability of the complements Ac, Bc as Ha=1 – Ha and
Hb=1 – Hb.
• The following duality relationships hold among copulas
Pr(A,B) = C(Ha,Hb)
Pr(Ac,B) = Hb – C(Ha,Hb) = Ca(Ha, Hb)
Pr(A,Bc) = Ha – C(Ha,Hb) = Cb(Ha,Hb)
Pr(Ac,Bc) =1 – Ha – Hb + C(Ha,Hb) = C(Ha, Hb) =
Survival copula
• Notice. This property of copulas is paramount to ensure
put-call parity relationships in option pricing applications.
Radial symmetry
• Take a copula function C(u,v) and its
survival version
C(1 – u, 1 – v) = 1 – v – u + C( u, v)
• A copula is said to be endowed with the
radial symmetry (reflection symmetry)
property if
C(u,v) = C(u, v)
AND/OR operators
• Copula theory also features more tools,
which are seldom mentioned in financial
applications.
• Example:
Co-copula = 1 – C(u,v)
Dual of a Copula = u + v – C(u,v)
• Meaning: while copula functions represent
the AND operator, the functions above
correspond to the OR operator.
Conditional probability I
• The dualities above may be used to
recover the conditional probability of the
events.
Pr H a  u, H b  v  C u, v 
Pr H a  u H b  v  

Pr H b  v 
v
Conditional probability II
• The conditional probability of X given Y = y can
be expressed using the partial derivative of a
copula function.
C u, v 
Pr  X  x Y  y  
v u F1  x ,v  F2  y 
Tail dependence in crashes…
• Copula functions may be used to compute an
index of tail dependence assessing the
evidence of simultaneous booms and crashes
on different markets
• In the case of crashes…
L v   Pr FX  v FY  v  
Pr FX  v, FY  v  C v, v 


Pr FY  v 
v
…and in booms
• In the case of booms, we have instead
U v   Pr FX  v FY  v  
Pr FX  v, FY  v  1  2v  C v, v 


Pr FY  v 
1 v
• It is easy to check that C(u,v) = uv leads to
lower and upper tail dependence equal to
zero. C(u,v) = min(u,v) yields instead tail
indexes equal to 1.
The Fréchet family
• C(x,y) =bCmin +(1 – a – b)Cind + aCmax , a,b [0,1]
Cmin= max (x + y – 1,0), Cind= xy, Cmax= min(x,y)
• The parameters a,b are linked to non-parametric
dependence measures by particularly simple
analytical formulas. For example
S = a  b
• Mixture copulas (Li, 2000) are a particular case in
which copula is a linear combination of Cmax and
Cind for positive dependent risks (a>0, b 0, Cmin
and Cind for the negative dependent (b>0, a 0.
Elliptical copulas
• Ellictal multivariate distributions, such as multivariate
normal or Student t, can be used as copula functions.
• Normal copulas are obtained
C(u1,… un ) =
= N(N – 1 (u1 ), N – 1 (u2 ), …, N – 1 (uN ); )
and extreme events are indipendent.
• For Student t copula functions with v degrees of freedom
C (u1,… un ) =
= T(T – 1 (u1 ), T – 1 (u2 ), …, T – 1 (uN ); , v)
extreme events are dependent, and the tail dependence
index is a function of v.
Archimedean copulas
• Archimedean copulas are build from a suitable
generating function  from which we compute
C(u1,…, un) =  – 1 [(u1)+…+(un)]
• The function (x) must have precise properties.
Obviously, it must be (1) = 0. Furthermore, it must be
decreasing and convex. As for (0), if it is infinite the
generator is said strict.
• In n dimension a simple rule is to select the inverse of
the generator as a completely monotone function
(infinitely differentiable and with derivatives alternate in
sign). This identifies the class of Laplace transform.
Example: Clayton copula
• Take
(t) = [t – – 1]/ 
such that the inverse is
 – 1(s) =(1 – s) – 1/ 
the Laplace transform of the gamma distribution.
Then, the copula function
C(u1,…, un) =  – 1 [(u1)+…+(un)]
is called Clayton copula. It is not symmetric and has
lower tail dependence (no upper tail dependence).
Example: Gumbel copula
• Take
(t) = (–log t)
such that the inverse is
 – 1(s) =exp(– s – 1/ )
the Laplace transform of the positive stable distribution.
Then, the copula function
C(u1,…, un) =  – 1 [(u1)+…+(un)]
is called Gumbel copula. It is not symmetric and has upper
tail dependence (no lower tail dependence).
Radial symmetry: example
• Take u = v = 20%. Take the gaussian copula and
compute N(u,v; 0,3) = 0,06614
• Verify that:
N(1 – u, 1 – v; 0,3) = 0,66614 =
= 1 – u – v + N(u,v; 0,3)
• Try now the Clayton copula and compute
Clayton(u, v; 0,2792) = 0,06614 and verify that
Clayton(1 – u, 1 – v; 0,2792) = 0,6484  0,66614
Kendall function
• For the class of Archimedean copulas, there is a
multivariate version of the probability integral
transfomation theorem.
• The probability t = C(u,v) is distributed according to the
distribution
KC (t) = t – (t)/ ’(t)
where ’(t) is the derivative of the generating function.
There exist extensions of the Kendall function to n
dimensions.
• Constructing the empirical version of the Kendall function
enables to test the goodness of fit of a copula function
(Genest and McKay, 1986).
Kendall function: Clayton copula
1,2
1
0,8
0,6
0,4
0,2
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
Copula product
• The product of a copula has been defined
(Darsow, Nguyen and Olsen, 1992) as
Au, t  Bt , v 
dt
A*B(u,v)  
t
t
0
1
and it may be proved that it is also a
copula.
Markov processes and copulas
• Darsow, Nguyen and Olsen, 1992 prove that 1st
order Markov processes (see Ibragimov, 2005
for extensions to k order processes) can be
represented by the  operator (similar to the
product)
A (u1, u2,…, un) B(un,un+1,…, un+k–1) 
Au1 , u2 ,..., un1 , t  Bt , un1 , um 2 ,..., um k 1 
dt
0
t
t
un
Properties of  products
• Say A, B and C are copulas, for simplicity
bivariate, A survival copula of A, B survival
copula of B, set M = min(u,v) and  = u v
• (A  B)  C = A  (B  C) (Darsow et al. 1992)
• A M = A, B M = B
(Darsow et al. 1992)
• A  = B  = 
(Darsow et al. 1992)
• A  B =A  B
(Cherubini Romagnoli, 2010)
Symmetric Markov processes
• Definition. A Markov process is symmetric if
1. Marginal distributions are symmetric
2. The  product
T1,2(u1, u2)  T2,3(u2,u3)…  Tj – 1,j(uj –1 , uj)
is radially symmetric
• Theorem. A  B is radially simmetric if either i)
A and B are radially symmetric, or ii) A  B = A
 A with A exchangeable and A survival
copula of A.
Example: Brownian Copula
• Among other examples, Darsow, Nguyen
and Olsen give the brownian copula
 t  1 v   s  1 w 
dw
0  
ts

u
If the marginal distributions are standard
normal this yields a standard browian
motion. We can however use different
marginals preserving brownian dynamics.
Basic credit risk applications
• Guarantee: assume a client has default
probability 20% and he asks for guarantee from
a guarantor with default probability of 1%. What
is the default probability of the loan?
• First to default: you buy protection on a first to
default on a basket of “names”. What is the
price? Are you long or short correlation?
• Last to default: you buy protection on the last
default in a basket of “names”. What is the
price? Are you long or short correlation?
An example: guarantee on credit
• Assume a credit exposure with probability of default of
Ha = 20% in a year.
• Say the credit exposure is guaranteed by another party
with default probability equal to Ha = 1% .
• The probability of default on the exposure is now the
joint probability
DP = C(Ha , Hb)
• The worst case is perfect dependence between default
of the two counterparties leading to
DP = min(Ha , Hb )
“First-to-default” derivatives
• Consider a credit derivative, that is a contract
providing “protection” the first time that an
element in the basket of obligsations defaults.
Assume the protection is extended up to time T.
• The value of the derivative is
FTD = LGD v(t,T)(1 – Q(0))
• Q(0) is the survival probability of all the names in
the basket:
Q(0) Q(1 > T, 2 > T…)