Slides euria-2

Arthur CHARPENTIER - École d'été EURIA.

mesures de risques et dépendance
Arthur Charpentier
Université de Rennes 1 & École Polytechnique
arthur.charpentier@univ-rennes1.fr
http://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/

1


3 0.9

V (rank of Y)
Y

0.4
-1
(X i,Y i)

(U i,V i)

-3 -1 1 3 0.2 0.5 0.8
X U (rank of X)

Density of the copula

Isodensity curves of the density

2


Agenda
• General introduction

Modelling correlated risks
• A short introduction to copulas

• Quantifying dependence

• Statistical inference

• Agregation properties

3


Agenda





4


Some references on large and correlated risks
Rank , J. (2006). Copulas: From Theory to Application in Finance. Risk Book ,

Nelsen , R. (1999,2006). An introduction to copulas. Springer Verlag ,

Cherubini , U., Luciano , E. & Vecchiato, W. (2004). Copula Methods in

Finance. Wiley,

Beirlant , J., Goegebeur , Y., Segers, J. & Teugels
, J. (2004). Statistics of

Extremes: Theory and Applications. Wiley,

McNeil , A. Frey , R., & Embrechts , P. (2005). Quantitative Risk

Management: Concepts, Techniques, and Tools. Princeton University Press,

5


Agenda





6


Copulas, an introduction (in dimension 2)
Denition 1. A copula C is a joint distribution function on [0, 1]2 , with uniform
margins on [0, 1].
Set C(u, v) = P(U ≤ u, V ≤ v), where (U, V ) is a random pair with uniform

margins.

C is a distribution function on [0, 1]2 , and thus C(0, v) = C(u, 0) = 0, C(1, 1) = 1.
Furthermore C is increasing : since P is a positive measure, for all u1 ≤ u2 and

v1 ≤ v2 ,
Copula, positive area

1.0
P(u1 U ≤ u2 , v1 V ≤ v2 ) ≥ 0,

0.8
0.6
thus
0.4
C(u2 , v2 ) − C(u1 , v2 )
0.2

−C(u2 , v1 ) + C(u1 , v1 ) ≥ 0.
0.0

0.0 0.2 0.4 0.6 0.8 1.0

7


C has uniform margins, and thus

C(u, 1) = P(U ≤ u, V ≤ 1) = P(U ≤ u) = u on [0, 1].

Proposition 2. C is a copula if and only if C(0, v) = C(u, 0) = 0, C(u, 1) = u
and C(1, v) = v for all u, v , with the following 2-increasingness property
C(u2 , v2 ) − C(u1 , v2 ) − C(u2 , v1 ) + C(u1 , v1 ) ≥ 0,

for any u1 ≤ u2 and v1 ≤ v2 .

8


Borders of the copula function

!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.2
1.0
0.8
0.6
0.4
0.2
0.0
!0.2
!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 1: Value of the copula on the border of the unit square.

9


Fonction de répartition à marges uniformes

Z

Y
X

Figure 2: Graphical representation of a copula.

10


If C is twice dierentiable, one can dene its density as

∂ 2 C(u, v)
c(u, v) = .
∂u∂v

11


Densité d’une loi à marges uniformes

z

x x

Figure 3: Density of a copula.

12


Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes


Figure 4: Distribution functions and densities.

13




Figure 5: Distribution functions and densities.

14


Sklar's theorem
Theorem 3. (Sklar) Let C be a copula, and FX and FY two marginal
distributions, then F (x, y) = C(FX (x), FY (y)) is a bivariate distribution
function, with F ∈ F(FX , FY ).
Conversely, if F ∈ F(FX , FY ), there exists C such that
F (x, y) = C(FX (x), FY (y)). Further, if FX and FY are continuous, then C is
unique, and given by
C(u, v) = F (FX (u), FY (v)) for all (u, v) ∈ [0, 1] × [0, 1]
−1 −1

We will then dene the copula of F , or the copula of (X, Y ).
In that case, the copula of (X, Y ) is the distribution function of (FX (X), FY (Y )).
Proposition 4. If (X, Y ) has copula C , the copula of (g(X), h(Y )) is also C for
any increasing functions g and h.

15


Copulas, an introduction (in dimension d ≥ 2)
Denition 5. A copula C is a joint distribution function on [0, 1]d , with
uniform margins on [0, 1].
Let U = (U1 , ..., Ud ) denote a random pair with uniform margins.

C is a distribution function on [0, 1]d , and thus C(u) = 0 if ui = 0 for some

i ∈ {1, . . . , d}, and C(1) = 1.
Furthermore C satises some increasing property since P is a positive measure

(for all 0 ≤ u ≤ v ≤ 1, P(u U ≤ v) ≥ 0), thus

sign(z)C(z) ≥ 0,
z

where the sum is taken over all vertices of [u × v], and where sign(z) is +1 if

zi = ui for an even number of i (and −1 otherwise, see Figure 6). And nally C
has uniform margins, and thus

C(1, . . . , 1, ui , 1, . . . , 1) = ui on [0, 1].

16


Increasing functions in dimension 3

Figure 6: The notion of 3-increasing functions.

17


Theorem 6. (Sklar) Let C be a copula, and F1 , . . . , Fd be d marginal
distributions, then F (x) = C(F1 (x1 ), . . . , Fd (xd )) is a distribution function, with
F ∈ F(F1 , . . . , Fd ).
Conversely, if F ∈ F(F1 , . . . , Fd ), there exists C such that
F (x) = C(F1 (x1 ), . . . , Fd (xd )). Further, if the Fi 's are continuous, then C is
unique, and given by
C(u) = F (F1 (u1 ), . . . , Fd (ud )) for all (ui ) ∈ [0, 1]
−1 −1

We will then dene the copula of F , or the copula of X .
In that case, the copula of (X = (X1 , . . . , Xd ) is the distribution function of

U = (F1 (X1 ), . . . , Fd (Yd )).
Again, if C is dierentiable, one can dene its density,

∂ d C(u1 , . . . , ud )
c(u1 , . . . , ud ) = .
∂u1 . . . ∂ud

18


Copulas in high dimension, a dicult problem
It is usually dicult to represent dependence in dimension d 2, and it is

usually studied by pairs.

In dimension d = 2, one can dene the following Fréchet class F(FX , FY , FZ )
dened by its marginal distributions. But it can also be interested to study

F(FXY , FXZ , FY Z ) dened by it paired distributions.

One of the problem that arises is the compatibility of marginals: one has to

verify that

CXY (x, y) = CX|Z (x|z)CY |Z (y|z)dz,

for instance.

19


1.0
0.8
1.0

0.6
0.8

p
0.6

0.4
1.0
0.4

0.8

0.2
0.6
0.2

0.4
0.2

0.0
0.0

0.0
0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Composante 1

1.0
1.0

0.8
0.8

0.6
0.6
p

p

0.4
0.4
0.2

0.2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0

Composante 2 Composante 3

Figure 7: Scatterplot in dimension 3 including projections.

20


Copulas and ranks
The copula of X = (X1 , . . . , Xd ) is the distribution function of

U = (F1 (X1 ), . . . , Fd (Yd )).
In practice, since marginal distributions are unknown, the idea is to substitute

empirical distribution function,

n
#{observations Xi,j 's lower than xi } 1
Fi (xi ) = = 1(Xi,j ≤ xi ).
#{observations } n j=1

Note that
n
#{observations Xi,j 's lower than Xi,j0 } 1 Ri,j0
Fi (Xi,j0 ) = = 1(Xi,j ≤ Xi,j0 ) = ,
#{observations } n j=1
n

where Ri,j0 denotes the rank of Xi,j0 within {Xi,1 , ..., Xi,n }.
On a statistical point of view, studying the copula means studying ranks.

21


Scatterplot of (X,Y) Scatterplot of the ranks of (X,Y)

9.0

20
8.5
8.0

15
Ranks of the Yi’s
7.5
Y (raw data)

10
7.0
6.5

5
6.0
5.5

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 5 10 15 20

X (raw data) Ranks of the Xi’s

Scatterplot of the ranks of (X,Y), divided by n Scatterplot o+ ,-,/0, t1e copula!t3pe tran+orm o+ ,6,70
1.0

1.0
0.8
0.8

Vi=Ranks of the Yi’s/n+1
Ranks of the Yi’s/n

0.6
0.6

0.4
0.4

0.2
0.2

0.0
0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Ranks of the Xi’s/n Ui=Ranks of the Xi’s/n+1

Figure 8: Copulas, ranks and parametric inference, from (Xi , Yi ) to (Ui , Vi ).

22


Some very classical copulas
• The independent copula C(u, v) = uv = C ⊥ (u, v).
The copula is standardly denoted Π, P or C ⊥, and an independent version of
L
(X, Y ) will be denoted (X ⊥ , Y ⊥ ). It is a random vector such that X⊥ = X and
L
Y⊥ =Y, with copula C ⊥.
In higher dimension, C ⊥ (u1 , . . . , ud ) = u1 × . . . × ud is the independent copula.

• The comonotonic copula C(u, v) = min{u, v} = C + (u, v).
The copula is standardly denoted M, or C +, and an comonotone version of
L
(X, Y ) will be denoted (X + , Y + ). It is a random vector such that X+ = X and
L
Y+ =Y, with copula C +.
(X, Y ) has copula C+ if and only if there exists a strictly increasing function h
L−1 −1
such that Y = h(X), or equivalently (X, Y ) = (FX (U ), FY (U )) where U is

U([0, 1]).

23


Note that for any u, v

P(U ≤ u, V ≤ v) = P({U ∈ [0, u]} ∩ {V ∈ [0, v]})
≤ min{P(U ∈ [0, u]), P(V ∈ [0, v])}

thus, C(u, v) ≤ min{u, v} = C + (u, v). Thus, C+ is an upper bound for the set of

copulas.

In higher dimension, C + (u1 , . . . , ud ) = min{u1 , . . . , ud } is the comonotonic

copula.

• The contercomotonic copula C(u, v) = max{u + v − 1, 0} = C − (u, v).
The copula is standardly denoted W, or C −, and an contercomontone version of
L
(X, Y ) will be denoted (X − , Y − ). It is a random vector such that X− = X and
L
Y− =Y, with copula C −.
(X, Y ) has copula C− if and only if there exists a strictly decreasing function h
L−1 −1
such that Y = h(X), or equivalently (X, Y ) = (FX (1 − U ), FY (U )) where U is

U([0, 1]).

24


Note that for any u, v ,

P(U ≤ u, V ≤ v) = P({U ∈ [0, u]} ∩ {V ∈ [0, v]})
= P(U ∈ [0, u]) + P(V ∈ [0, v]) − P({U ∈ [0, u]} ∪ {V ∈ [0, v]})

thus, C(u, v) ≥ u + v − 1 since P({U ∈ [0, u]} ∪ {V ∈ [0, v]}) ≤ 1, and since
C(u, v) ≥ 0, C(u, v) ≥ max{u + v − 1, 0} = C − (u, v). Thus, C − is a lower bound
for the set of copulas.

In higher dimension, C − (u1 , . . . , ud ) = max{u1 + . . . + ud − (d − 1), 0} is not a

copula: if (X, Y ) and (X, Z) are countercomonotonic, (Y, Z) is necessarily
comonotonic - it is not possible to have all component highly negatively

correlated.

Anyway, it is still the best pointwise lower bound.

25


1

1

0.8 1
nd
0.8

0.2 0.4 0.6 0.8
la

nd
Independence copu
Frechet lower bou

Frechet upper bou
0 0.2 0.4 0.6

0 0.2 0.4 0.6
0
0.8 0.8
0.8
0.6 0.6
0.8 0.8 0.6
0.8
u_ 0.4 0.6 u_ 0.4 0.6
2 2 u_ 0.4 0.6
2
0.4 0.4 u_1
0.2 u_1 0.2
0.2 0.4
u_1
0.2 0.2
0.2

Fréchet Lower Bound Independent copula Fréchet Upper Bound
1.0

1.0

1.0
0.8

0.8

0.8
0.6

0.6

0.6
0.4

0.4

0.4
0.2

0.2

0.2
0.0

0.0

0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Scatterplot, Lower Fréchet!Hoeffding bound Scatterplot, Indepedent copula random generation Scatterplot, Upper Fréchet!Hoeffding bound
1.0

1.0

1.0
0.8

0.8

0.8
0.6

0.6

0.6
0.4

0.4

0.4
0.2

0.2

0.2
0.0

0.0

0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Figure 9: Contercomontonce, independent, and comonotone copulas.

26


Pitfalls on independence and comonotonicity
The following proposition is false,

Uncorrect Proposition 7. If X and Y are independent, if Y and Z are
independent, then X and Z are independent.
If

(X, Y, Z) = (1, 1, 1) with probability 1/4,
(1, 2, 1) with probability 1/4,

then X and Y are independent, and Y and Z are independent, but X = Z.

27


X and Y independent Y and Z independent X and Z comonotonic
4

4

4
3

3

3
Component Y

Component Z

Component Z
2

2

2
1

1

1
0

0

0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

Component X Component Y Component X

Figure 10: Mixing independence and comonotonicity.

28



Uncorrect Proposition 8. If X and Y are comonotonic, if Y and Z are
comonotonic, then X and Z are comonotonic.
If


then X and Y are comonotonic, and Y and Z are comonotonic, but X and Z are

independent.

29


X and Y comonotonic Y and Z comonotonic X and Z independent
4

4

4
3

3

3
Component Y

Component Z

Component Z
2

2

2
1

1

1
0

0

0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4



30



Uncorrect Proposition 9. If X and Y are comonotonic, if Y and Z are
independent, then X and Z are independent.
If


then X and Y are comonotonic, and Y and Z are independent, but X and Z are

anticomonotonic.

31


If


then X and Y are comonotonic, and Y and Z are independent, but X and Z are

comonotonic.

32


X and Y comonotonic Y and Z independent X and Z comonotonic
4

4

4
3

3

3
Component Y

Component Z

Component Z
2

2

2
1

1

1
0

0

0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4



33


Elliptical (Gaussian and t) copulas
The idea is to extend the multivariate probit model, Y = (Y1 , . . . , Yd ) with
marginal B(pi ) distributions, modeled as Yi = 1(Xi ≤ ui ), where X ∼ N (I, Σ).
• The Gaussian copula, with parameter α ∈ (−1, 1),
Φ−1 (u) Φ−1 (v)
1 −(x2 − 2αxy + y 2 )
C(u, v) = √ exp dxdy.
2π 1 − α2 −∞ −∞ 2(1 − α2 )

Analogously the t-copula is the distribution of (T (X), T (Y )) where T is the t-cdf,
and where (X, Y ) has a joint t-distribution.
• The Student t-copula with parameter α ∈ (−1, 1) and ν ≥ 2,
t−1 (u) t−1 (v) −((ν+2)/2)
1 ν ν x2 − 2αxy + y 2
C(u, v) = √ 1+ dxdy.
2π 1 − α2 −∞ −∞ 2(1 − α2 )

34


Archimedean copulas
Denition of Archimedean copulas

• Archimedian copulas C(u, v) = φ−1 (φ(u) + φ(v)), where φ is decreasing

convex (0, 1), with φ(0) = ∞ and φ(1) = 0.
Example 10. If φ(t) = [− log t]α , then C is Gumbel's copula, and if
φ(t) = t−α − 1, C is Clayton's. Note that C ⊥ is obtained when φ(t) = − log t.
How Archimedean copulas were introduced ?

1. The frailty approach ( Oakes (1989)).

Assume that X and Y are conditionally independent, given the value of an

heterogeneous component Θ. Assume further that

P(X ≤ x|Θ = θ) = (GX (x))θ and P(Y ≤ y|Θ = θ) = (GY (y))θ

for some baseline distribution functions GX and GY .
Then

F (x, y) = P(X ≤ x, Y ≤ y) = E(P(X ≤ x, Y ≤ y|Θ = θ))

35


thus, since X and Y are conditionally independent,

F (x, y) = E(P(X ≤ x|Θ = θ) × P(Y ≤ y|Θ = θ))

and therefore

F (x, y) = E (GX (x))Θ × (GY (y))Θ = ψ(− log GX (x) − log GY (y))

where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ ). Since

FX (x) = ψ(− log GX (x)) and FY (y) = ψ(− log GY (y))

and thus, the joint distribution of (X, Y ) satises

F (x, y) = ψ(ψ −1 (FX (x)) + ψ −1 (FY (y))).

Example 11. If Θ is Gamma distributed, the associated copula is Clayton's. If
Θ has a stable distribution, the associated copula is Gumbel's.

36


Consider two risks, X and Y, such that

X|Θ = θG ∼ E(θG ) and Y |Θ = θG ∼ E(θG ) are independent,

X|Θ = θB ∼ E(θB ) and Y |Θ = θB ∼ E(θB ) are independent,

(unobservable good (G) and bad (B ) risks).

The following gures start from 2 classes of risks, then 3, and then a continuous

risk factor θ ∈ (0, ∞).

37


Conditional independence, two classes Conditional independence, two classes

20

3
2
15

1
10

0
!1
5

!2
!3
0

0 5 10 15 !3 !2 !1 0 1 2 3

Figure 13: Two classes of risks, (Xi , Yi ) and (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

38


Conditional independence, three classes Conditional independence, three classes

3
40

2
30

1
0
20

!1
10

!2
!3
0

0 5 10 15 20 25 30 !3 !2 !1 0 1 2 3

Figure 14: Three classes of risks, (Xi , Yi ) and (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

39


Conditional independence, continuous risk factor Conditional independence, continuous risk factor

100

3
2
80

1
60

0
40

!1
20

!2
!3
0

0 20 40 60 80 100 !3 !2 !1 0 1 2 3

Figure 15: Continuous classes of risks, (Xi , Yi ) and (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

40


2. The survival approach: assume that there is a convex survival function S,
with S(0) = 1, such that

P(X x, Y y) = S(x + y),

then the joint survival copula of (X, Y ) is

S(S −1 (u) + S −1 (v)).

Example 12. If S is the Pareto survival distribution, the associated copula is
Clayton's. If S is the Weibull survival distribution, the associated copula is
Gumbel's.

41


3. The use of Kendall's distribution function K(t) = P(C(U, V ) ≤ t) where

(U, V ) is a random pair with distribution function C .

Then, for Archimedean copulas,

φ (t)
K(t) = t − = t − λ(t),
φ(t)
which can be inverted easily in

1
1
φ(t) = φ(t0 ) exp dt ,
t0 λ(t)
for some 0 t0 1 and 0 ≤ u ≤ 1.

42


Some more examples of Archimedean copulas

ψ(t) range θ
(1) 1 (t−θ − 1)
θ
[−1, 0) ∪ (0, ∞) Clayton, Clayton (1978)

(2) (1 − t)θ [1, ∞)
1−θ(1−t)
(3) log [−1, 1) Ali-Mikhail-Haq

Gumbel Hougaard
t
(4) (− log t)θ [1, ∞) Gumbel, (1960), (1986)

(5) − log e
−θt −1
e−θ −1
(−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen(1987)

(6) − log{1 − (1 − t)θ } [1, ∞) Joe, Frank (1981), Joe(1993)

(7) − log{θt + (1 − θ)} (0, 1]
(8)
1−t [1, ∞)
1+(θ−1)t
(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)

(10) log(2t−θ − 1) (0, 1]
(11) log(2 − tθ ) (0, 1/2]
(12) ( 1 − 1)θ [1, ∞)
t
(13) (1 − log t)θ − 1 (0, ∞)
(14) (t−1/θ − 1)θ [1, ∞)
(15) (1 − t1/θ )θ [1, ∞) Genest Ghoudi (1994)

(16) ( θ + 1)(1 − t) [0, ∞)
t

43


Some characterizations of Archimedean copula

L
• Frank copula is the only Archimedean such that (U, V ) = (1 − U, 1 − V )
(stability by symmetry),

• Clayton copula is the only Archimedean such that (U, V ) has the same

copula as (U, V ) given (U ≤ u, V ≤ v) (stability by truncature),

• Gumbel copula is the only Archimedean such that (U, V ) has the same
copula as (max{U1 , ..., Un }, max{V1 , ..., Vn }) for all n ≥ 1 (max-stability),

44


Extreme value copulas
• Extreme value copulas

log u
C(u, v) = exp (log u + log v) A ,
log u + log v
where A is a dependence function, convex on [0, 1] with A(0) = A(1) = 1, et

max{1 − ω, ω} ≤ A (ω) ≤ 1 for all ω ∈ [0, 1] .

An alternative denition is the following: C is an extreme value copula if for all

z 0,
1/z 1/z
C(u1 , . . . , ud ) = C(u1 , . . . , ud )z .

Those copula are then called max-stable: dene the maximum componentwise of

a sample X 1 , . . . , Xn , i.e. Mi = max{Xi,1 , . . . , Xi,n }.

45


The joint distribution of M is

P(M ≤ x) = C(F1 (x1 , . . . , Fd (xd ))n ,

where C is the copula of the X i 's. Since P(Mi ≤ xi ) = Fi (xi )n , it can be written

P(M ≤ x) = C(P(M1 ≤ x1 )1/n , . . . , P(Md ≤ xd )1/n )n .
1/n 1/n
Thus, C(u1 , . . . , ud )n is the copula of the n maximum componentwise from a

sample with copula C .

Example 13. : If A is constant (1 on [0, 1]), then X and Y are independent,
and if A(ω) = max {ω, 1 − ω}, X and Y are comonotonic. Gumbel's copula is
obtained if
(
A(ω) = ((1 − ω)α + ω α + 1) 1/α),
for all 0 ≤ ω ≤ 1 and α ≥ 1.

46


Pickands dependence function A

1.0
0.9
0.8
0.7
0.6
0.5

0.0 0.2 0.4 0.6 0.8 1.0

Figure 16: Shape of Gumbel's dependence function A(ω).

47


How to construct much more copulas ?
Using geometric transformations

From a given copula C, cdf of random pair (U, V ), dene

• the copula of (U, 1 − V ),

C(U,1−V ) (u, v) = u − C(u, 1 − v)

• the copula of (1 − U, V ),

C(1−U,V ) (u, v) = v − C(1 − u, v)

• the copula of (1 − U, 1 − V ), the rotated or survival copula,
C(1−U,1−V ) (u, v) = C ∗ (u, v) = u + v − 1 + C(1 − u, 1 − v)

Note that if P(X ≤ x, Y ≤ y) = C(P(X ≤ x), P(Y ≤ y)), then

P(X x, Y y) = C ∗ (P(X x), P(Y y)).

48


0.8
0.8

0.4
0.4

0.0
0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Figure 17: Using geometric transformation to generate new copulas.

49


0.8

0.8
0.4

0.4
0.0
0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


50


0.8
0.8

0.4
0.4

0.0
0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


51


0.8

0.8
0.4

0.4
0.0
0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0


52


Using mixture of copulas

Lemma 14. The set of copulas is convex, i.e. if {Cθ , θ ∈ Ω} is a collection of
copulas,
C(u, v) = Cθ (u, v)dΠ(θ)
R
is a copula, where Π is a distribution on Ω
Thus C = αC1 + (1 − α)C2 denes a copula for all α ∈ [0, 1].
Example 15. Fréchet (1951) suggested a mixture of the lower and the upper
bound,
C(u, v) = αC − (u, v) + (1 − α)C + (u, v), for some α ∈ [0, 1].

Example 16. Mardia (1970) suggested a mixture of the lower, the upper
bound, and the independent copula
α2 − 2 ⊥ α2 +
C(u, v) = C (u, v) + (1 − α )C (u, v) + C (u, v), α ∈ [0, 1].
2 2

53


Using distortion functions

Denition 17. A distortion function is a function h : [0, 1] → [0, 1] strictly
increasing such that h(0) = 0 and h(1) = 1.
The set of distortion function will be denoted H.
Note that h∈H if and only if h−1 ∈ H. Given a copula C, dene

Ch (u, v) = h−1 (C(h(u), h(v))).

If h is convex, then Ch is a copula, called distorted copula.

Example 18. if h(x) = x1/n , the distorted copula is
Ch (u, v) = C n (u1/n , v 1/n ), for all n ∈ N, (u, v) ∈ [0, 1]2 .

if the survival copula of the (Xi , Yi )'s is C , then the survival copula of
(Xn:n , Yn:n ) = (max{X1 , ..., Xn }, max{Y1 , ..., Yn }) is Ch .

54


Example 19. if C(u, v) = uv = C ⊥ (u, v) (the independent copula), and
φ(·) = log h(·), then

Ch (u, v) = h−1 (h(u)h(v)) = φ−1 (φ(u) + φ(v)).

Example 20. if h(x) = [1 − e−αx ]/[1 − e−α ] (an exponential distortion), and if
C = C ⊥ , then
1 (e−αu − 1)(e−αv − 1)
Ch (u, v) = − log 1 + ,
α e−α − 1
which is Frank copula.

55


Distorted Frank copula, h(x) = x Distorted Frank copula, h(x) = x(1 2)

Distorted Frank copula, h(x) = x(1 3)
Distorted Frank copula, h(x) = x(1 4)

Figure 21: Distorted copula, from Frank copula.

56


Monte Carlo and copulas
Generation of independent variables can be done using a Random function.

Denition 21. Function Random should satisfy the following properties (i) for
all 0 ≤ a ≤ b ≤ 1,
P (Random ∈ ]a, b]) = b − a.
(ii) successive calls of function Random should generate independent draws, i.e.
0 ≤ a ≤ b ≤ 1, 0 ≤ c ≤ d ≤ 1

P (Random1 ∈ ]a, b] , Random2 ∈ ]c, d]) = (b − a) (d − c) ,

or more generally, dene k-uniformity for all 0 ≤ ai ≤ bi ≤ 1, i = 1, ..., k,
k
P (Random1 ∈ ]a1 , b1 ] , ..., Randomk ∈ ]ak , bk ]) = (bi − ai ) .
i=1

Thus, one can generate easily random vectors U = (U1 , ..., Ud ) with independent

component.

57


The idea to generate correlated vectors U = (U1 , ..., Ud ), the idea is to use rst

P(U1 ≤ u1 , . . . , Ud ≤ ud ) = P(Ud ≤ ud |U1 ≤ u1 , . . . , Ud−1 ≤ ud−1 )
×P(Ud−1 ≤ ud−1 |U1 ≤ u1 , . . . , Ud−2 ≤ ud−2 )
×...
×P(U3 ≤ u3 |U1 ≤ u1 , U2 ≤ u2 )
×P(U2 ≤ u2 |U1 ≤ u1 ) × P(U1 ≤ u1 ).

58


Starting from the end, P(U1 ≤ u1 ) = u1 since U1 is uniform, while

P(U2 ≤ u2 |U1 = u1 )
= P(U2 ≤ u2 , U3 ≤ 1, . . . Ud ≤ 1|U1 = u1 )
= lim P(U2 ≤ u2 , U3 ≤ 1, . . . Ud ≤ 1|U1 ∈ [u1 , u1 + h])
h→0
P(u1 ≤ U1 ≤ u1 + h, U2 ≤ u2 , U3 ≤ 1, . . . Ud ≤ 1)
= lim
h→0 P(U1 ∈ [u1 , u1 + h])
P(U1 ≤ u1 + h, U2 ≤ u2 , U3 ≤ 1, . . . Ud ≤ 1) − P(U1 ≤ u1 , U2 ≤ u2 , U3 ≤ 1, . . .
= lim
h→0 P(U1 ∈ [u1 , u1 + h])
C(u1 + h, u2 , 1, . . . , 1) − C(u1 , u2 , 1, . . . , 1) ∂C
= lim = C(u1 , u2 , 1, . . . , 1).
h→0 h ∂u1
and more generally,

∂ k−1
P(Uk ≤ uk |U1 = u1 , . . . , Uk−1 = uk−1 ) = C(u1 , . . . , uk , 1, . . . , 1).
∂u1 . . . ∂uk−1

59


Thus, U = (U1 , .., Un ) with copula C could be simulated using the following

algorithm,

• simulate U1 uniformly on [0, 1],

u1 ← Random1 ,

• simulate U2 from the conditional distribution ∂1 C(·|u1 ),

u2 ← [∂1 C(·|u1 )]−1 (Random2 ),

• simulate Uk from the conditional distribution ∂1,...,k−1 C(·|u1 , ..., uk−1 ),

uk ← [∂1,...,k−1 C(·|u1 , ..., uk−1 )]−1 (Randomk ),

...etc, where the Randomi 's are independent calls of a Random function.

This is the underlying idea when using Cholesky decomposition.

60


Example: for Clayton's copula, C(u, v) = (u−α + v −α − 1)−1/α , (U, V ) has joint

distribution C if and only if U is uniform on on [0, 1] and V |U = u has
conditional distribution

P(V ≤ v|U = u) = ∂2 C(v|u) = (1 + uα [v −α − 1])−1−1/α .

The algorithm to generate Clayton's copula is the

• simulate U1 uniformly on [0, 1],

u1 ← Random1 ,

• simulate U2 from the conditional distribution ∂2 C(·|u),

u2 ← [∂1 C(·|u1 )]−1 (Random2 ),

i.e.

u2 ← [(Random2 )−α/(1+α − 1]u−α + 1−1/α .
1

61


1.5
0.0 0.5 1.0 1.5 2.0

1.0
0.5
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Distribution of v given u=0.3 Distribution of v given u=0.5

Generation of Clayton’s copula
0.8

0.0 0.5 1.0 1.5
0.4
0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Distribution of v given u=0.8

Figure 22: Simulation of Clayton's copula.

62


500

400
/

/
300
q

q

200
0 100

0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

!i#tribution +e - !i#tribution +e 9

4
0.8

2
0
0.4

!4 !2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !4 !2 0 2 4

-ni:orm mar=in# Stan+ar+ ?au##ian mar=in#

Figure 23: Simulation of the independent copula.

63


400

400
/

/
.

.
200

200
0

0
010 012 014 014 015 610 010 012 014 014 015 610

!is$riu$i(n +e - !is$riu$i(n +e 7

4
015

2
0
014

!2
010

010 012 014 014 015 610 !2 0 2 4

-ni8(r9 9argins $an+ar+ =aussian 9argins

Figure 24: Simulation of the comontone copula.

64


400

400
/

/
q

q
200

200
0

0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Distribution de - Distribution de V
0.8

2
0
0.4

!2
0.0

!4
0.0 0.2 0.4 0.6 0.8 1.0 !2 0 2 4

-niform margins tandard =aussian margins

Figure 25: Simulation of the contercomonotone copula.

65


900
400
/

/

300
.

.
200

0 100
0

010 012 014 014 015 110 010 012 014 014 015 110

!istri'tion de - !istri'tion de 7

4
015

2
0
014

!4 !2
010

010 012 014 014 015 110 !4 !2 0 2 4

-ni:orm mr=ins Stndrd ?'ssin mr=ins

Figure 26: Simulation of the Gaussian copula.

66


400
400
y

y
.

.

200
200
0

0
010 012 014 014 015 110 010 012 014 014 015 110

D#$%u$on +e U D#$%u$on +e V

4
015

2
0
014

!2
010

010 012 014 014 015 110 !2 0 2 4

Unfo%m ma%;n# S$an+a%+ =au##an ma%;n#

Figure 27: Simulation of Clayton's copula.

67


400

400
y

y
q

q
200

200
0

0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Distribution de U Distribution de V

4
0.8

2
0
0.4

!4 !2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !4 !2 0 2 4

Uniform margins Standard Gaussian margins

Figure 28: Simulation of Clayton's survival copula.

68


400

400
y

y
q

q
200

200
0

0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Distribution de U Distribution de V

4
0.8

2
0
0.4

!4 !2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !4 !2 0 2

Uniform margins Standard Gaussian margins

Figure 29: Simulation of a copula mixture.

69


Copulas in nance: options on multiple assets
Remark 22. Recall that Breeden Litzenberger (1978) proved that the risk
neutral probability can be obtrained from option prices: consider the price of a call
∞
C(T, K) = e−rT EQ ((ST − K)+ ). Since (ST − K)+ = K 1(ST x)dx, one gets
∞
−rT
C(T, K) = e Q(ST x)dx,
K

hence
∂C ∂P
Q(ST ≤ x) = −e−rT (T, x), or Q(ST ≤ x) = −erT (T, x)
∂K ∂K
where P denotes the price of a put option.
1 2
Consider an option on 2 assets, with payo h(ST , ST ). The price at time 0 is

e−rT EQ (h(ST , ST )).
1 2

70


Copulas in nance: call on maximum
1 2 1 2
Here the payo is h(ST , ST ) = (max{ST , ST } − K)+ . The price is then

C(T, K) = e−rT EQ ((max{ST , ST } − K)+ )
1 2

∞
= e−rT EQ 1 2
1 − 1(max{ST , ST } ≤ x)dx
K
∞
−rT 1 2
= e 1 − Q(max{ST , ST } ≤ x) dx,
K
1 2
Q(ST ≤x,ST ≤x)

1 2
hence, if (ST , ST ) has copula C (under Q), then

∞
∂P 1 ∂P 2
C(T, K) = e−rT 1 − C erT (T, x), erT (T, x) dx.
K ∂K ∂K

71


Copulas in nance: call on spreads
1 2 1 2
Here the payo is h(ST , ST ) = ([ST − ST ] − K)+ . The price is then

∞
−rT 1 2 −rT 2 1
C(T, K) = e EQ ((ST − ST − K)+ ) = e EQ 1(ST + K ≤ x ≤ ST )dx
−∞
∞
= e−rT 2 2 1
Q(K + ST ≤ x) − Q(ST + K ≤ x, ST ≤ x} ≤ x) dx,
−∞
1 2
Q(ST ≤x,ST ≤x+K)

1 2
hence, if (ST , ST ) has copula C (under Q), then

∞
−rT rT ∂P 2 rT ∂P
1
rT ∂P
2
C(T, K) = e e (T, x−K)−C e (T, x), e (T, x − K) dx.
−∞ ∂K ∂K ∂K

72


Copulas in nance: bonds on option prices
Using Tchen's inequality, it is possible to derive bounds for options when the

payo is supermodular.

73


Agenda





74


Natural properties for dependence measures

Denition 23. κ is measure of concordance if and only if κ satises
1. κ is dened for every pair (X, Y ) of continuous random variables,
2. −1 ≤ κ (X, Y ) ≤ +1, κ (X, X) = +1 and κ (X, −X) = −1,
3. κ (X, Y ) = κ (Y, X),
4. if X and Y are independent, then κ (X, Y ) = 0,
5. κ (−X, Y ) = κ (X, −Y ) = −κ (X, Y ),
6. if (X1 , Y1 ) P QD (X2 , Y2 ), then κ (X1 , Y1 ) ≤ κ (X2 , Y2 ),
7. if (X1 , Y1 ) , (X2 , Y2 ) , ... is a sequence of continuous random vectors that
converge to a pair (X, Y ) then κ (Xn , Yn ) → κ (X, Y ) as n → ∞.

75


As pointed out in Scarsini (1984), most of the axioms are self-evident .
Ifκ is measure of concordance, then, if f and g are both strictly increasing, then
κ(f (X), g(Y )) = κ(X, Y ). Further, κ(X, Y ) = 1 if Y = f (X) with f almost
surely strictly increasing, and analogously κ(X, Y ) = −1 if Y = f (X) with f

almost surely strictly decreasing (see Scarsini (1984)).

76


Association measures: Kendall's τ and Spearman's ρ
Rank correlations can be considered, i.e. Spearman's ρ dened as

1 1
ρ(X, Y ) = corr(FX (X), FY (Y )) = 12 C(u, v)dudv − 3
0 0

and Kendall's τ dened as

1 1
τ (X, Y ) = 4 C(u, v)dC(u, v) − 1.
0 0

Historical version of those coecients

Spearman's rho was introduced in Spearman (1904) as

ρ(X, Y ) = 3[P((X1 − X2 )(Y1 − Y3 ) 0) − P((X1 − X2 )(Y1 − Y3 ) 0)],

where (X1 , Y1 ), (X2 , Y2 ) and (X3 , Y3 ) denote three independent versions of

(X, Y ) (see Nelsen (1999)).

77


Similarly Kendall's tau was not dened using copulae, but as the probability of

concordance, minus the probability of discordance, i.e.

τ (X, Y ) = 3[P((X1 − X2 )(Y1 − Y2 ) 0) − P((X1 − X2 )(Y1 − Y2 ) 0)],

where (X1 , Y1 ) and (X2 , Y2 ) denote two independent versions of (X, Y ) (see

Nelsen (1999)).

4Q
Equivalently, τ (X, Y ) = 1 − 2 − 1)
where Q is the number of inversions
n(n
between the rankings of X and Y (number of discordance).

78


1.5 Concordant pairs Discordant pairs

1.5
1.0

1.0
0.5

0.5
Y

Y
0.0

0.0
!0.5

!0.5
!2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0 !2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0

X X

Figure 30: Concordance versus discordance.

79


The case of the Gaussian random vector

If (X, Y ) is a Gaussian random vector with correlation r, Kruskal
then ( (1958))

6 r 2
ρ(X, Y ) = arcsin and τ (X, Y ) = arcsin (r) .
π 2 π

80


Link between Kendall's tau and Spearman's rho

Note that Kendall's tau and Spearman's are linked: it is impossible to have at

the same time τ ≥ 0.4 and ρ = 0.
Hence ρ and τ satisfy

3τ − 1 1 + 2τ − τ 2
≤ρ≤ if τ ≥0
2 2
τ 2 + 2τ − 1 1 + 3τ
≤ρ≤ if τ ≤ 0.
2 2
which yield the area given below.

81


1.0

0.5
Rho de Spearman

0.0

-0.5

-1.0
-1.0 -0.5 0.0 0.5 1.0
Tau de Kendall

Figure 31: Admissible region of ρ and τ.

82


From Kendall'tau to copula parameters
Kendall's τ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Gaussian θ 0.00 0.16 0.31 0.45 0.59 0.71 0.81 0.89 0.95 0.99 1.00

Gumbel θ 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 +∞
Plackett θ 1.00 1.57 2.48 4.00 6.60 11.4 21.1 44.1 115 530 +∞
Clayton θ 0.00 0.22 0.50 0.86 1.33 2.00 3.00 4.67 8.00 18.0 +∞
Frank θ 0.00 0.91 1.86 2.92 4.16 5.74 7.93 11.4 18.2 20.9 +∞
Joe θ 1.00 1.19 1.44 1.77 2.21 2.86 3.83 4.56 8.77 14.4 +∞
Galambos θ 0.00 0.34 0.51 0.70 0.95 1.28 1.79 2.62 4.29 9.30 +∞
Morgenstein θ 0.00 0.45 0.90 - - - - - - - -

83


From Spearman's rho to copula parameters
Spearman's ρ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Gaussian θ 0.00 0.10 0.21 0.31 0.42 0.52 0.62 0.72 0.81 0.91 1.00

Gumbel θ 1.00 1.07 1.16 1.26 1.38 1.54 1.75 2.07 2.58 3.73 +∞
A.M.H. θ 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 +∞
Plackett θ 1.00 1.35 1.84 2.52 3.54 5.12 7.76 12.7 24.2 66.1 +∞
Clayton θ 0.00 0.14 0.31 0.51 0.76 1.06 1.51 2.14 3.19 5.56 +∞
Frank θ 0.00 0.60 1.22 1.88 2.61 3.45 4.47 5.82 7.90 12.2 +∞
Joe θ 1.00 1.12 1.27 1.46 1.69 1.99 2.39 3.00 4.03 6.37 +∞
Galambos θ 0.00 0.28 0.40 0.51 0.65 0.81 1.03 1.34 1.86 3.01 +∞
Morgenstein θ 0.00 0.30 0.60 0.90 - - - - - - -

84


Alternative expressions of those coecients

Note that those coecients can also be expressed as follows

[0,1]×[0,1]
C(u, v) − C ⊥ (u, v)dudv
ρ(X, Y ) = (1)
[0,1]×[0,1]
C + (u, v) − C ⊥ (u, v)dudv

(the normalized average distance between C and C ⊥ ), for instance.

A dependence measure in higher dimension ?

From equations 1 and ??, it is possible to obtain a natural mutlidimensional
extention (see Wolf (1980), Joe (1990) or Nelsen (1996)),

[0,1]d
C(u) − C ⊥ (u)du d+1
ρ(X) = = 2d C(u)du − 1
[0,1]×[0,1]
C + (u) − C ⊥ (u)du 2d − (d + 1) [0,1]d
(2)

and similarly

1
τ (X) == d−1 2d C(u)dCu − 1 (3)
2 − 1) [0,1]d

85


Note that a lower bound for τ is then −1/(2d−1 − 1), while it is

(2d − (d + 1)!)/(d!(2d − (d + 1))).
In dimension 3, Kendall's τ is the average of the three 2-dimensional Kendall's τ,
1
τ (X, Y, Z) = (τ (X, Y ) + τ (X, Z) + τ (Y, Z)).
3

86


Tail concentration functions
Venter (2002) suggest to use several Tail Concentration Functions

Denition 24. For lower tails, dene
L(z) = P(U z, V z)/z = C(z, z)/z = P r(U z|V z) = P r(V z|U z),

and for upper tails,
R(z) = P(U z, V z)/(1 − z) = P r(U z|V z).

Joe (1990) uses the term upper tail dependence parameter for

R = R(1) = limz→1 R(z), and lower tail dependence parameter for

L = L(0) = limz→0 L(z).

87


Functional correlation measures
1 1
Consider also Kendall's tau, dened as −1 + 4 0 0
C(u, v)dC(u, v).
Denition 25. The cumulative tau can be dened as
z z
J(z) = −1 + 4 C(u, v)dC(u, v)/C(z, z)2 .
0 0

88

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