Slides p6

Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?

Copules, avenir de l'actuariat

ou gadget statistique?

Arthur Charpentier
http://perso.univ-rennes1.fr/arthur.charpentier/

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... de la théorie à la pratique, considérations générales
Le paradoxe de Saint Petersbourg: le prix d'un jeu est le produit scalaire des
probabilités et des gains
n
p, x = pi xi = EP (X),
i=1

i.e. l'espérance mathématique est donc le juste prix des chances (Cournot A.
A. (1843). Exposition de la théorie des chances et des probabilités).

• réponse de Bernoulli, introduire une utilité morale de l'argent, i.e. distortion

sur les gains, Bernoulli, D. (1738). Exposition of a New Theory on the

Measurement of Risk. Econometrica, 22, 23-36.
• réponse de d'Alembert, introduire une distorsion sur les probabilités,

D’Alembert, J. (1768). Sur l'analyse des jeux. Opuscules mathématiques,
4, 80-92.

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... de la théorie à la pratique, considérations générales
Problème d'allocation d'actifs et de frontière d'ecience, deux articles en 1956:

• approche moyenne-variance, Markowitz, H. (1956). Portfolio Selection.

Journal of Finance 7, 1, 77-91.
• approche moyenne-quantile, Roy, A.D. (1956). Safety First and the Holding

of Assets. Econometrica, 20, 3, 431-449.

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le problème de la dépendance... les lois elliptiques
première piste pour générer des lois multivariés dépendentes: les lois elliptiques,

extension de la loi normale multivariée.

1. on partir de deux vecteurs X et Y N (0, 1) indépendants

2. on fait subir une homothétie et une rotation

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le problème de la dépendance... les lois elliptiques
idée: si X = µ + AX 0 où X 0 ∼ N (0, 1), alors X ∼ N (µ, Σ = A A).
on passe ainsi de vecteurs sphériques aux vecteurs elliptiques.

marche pour le vecteur Student t multivarié.

permet de modéliser d'autres types de lois, e.g. des lois binomiales multivariées:

modèle probit multivarié (cf CreditMetrics)

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le problème de la dépendance... les mélanges
deuxième piste pour générer des lois multivariés dépendentes: l'indépendance

conditionnelle et les modèles de frailty.

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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 1: Deux classes de risques, (Xi , Yi ) et (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

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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 2: Trois classes de risques, (Xi , Yi ) et (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

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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 3: Classes de risques continues, (Xi , Yi ) et (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

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le problème de la dépendance... les chocs communs
troisième piste pour générer des lois multivariés dépendentes: les modèles à choc

commun

premier exemple, la loi de Poisson Soient X, Y, Z trois variables P(λX ),
P(λY ) et P(λZ ) independentes. On pose alors

U = X + Z ∼ P(λX + λZ ) et Z = Y + Z ∼ P(λY + λZ )

deuxième exemple, la loi exponentielle Soient X, Y, Z trois variables E(µX ),
E(µY ) et E(µZ ) independentes. On pose alors

U = min{X, Z} et V = min{Y, Z}

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comment comparer ces diérentes dépendances
Pour comparer les structures de dépendance, il faut que les lois marginales soient

identiques,

• deux lois uniformes sur [0, 1]
• deux lois normales centrées réduites N (0, 1)
Si X a pour fonction de répartition F continue,F (x) = P(X ≤ x), alors F (X)
suit une loi uniforme sur [0, 1]. Aussi, Φ−1 (F (X)) suit une loi N (0, 1).

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comment comparer ces diérentes dépendances ?

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0.4
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−2

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0.2
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−4

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0.0
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−3 −2 −1 0 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4: Le modèle Gaussien, N (0, 1) et U([0, 1]).

17



1.0
q qqqqq
qqqq
q qq qq
qq
q q
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qqq qqqqqqqq
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q qq qqqqqq qq
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3

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0.8
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1

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0.6
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0

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0.4
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−1

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Figure 5: Le modèle Student t, N (0, 1) et U([0, 1]).

18



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Figure 6: Le modèle Student t, N (0, 1) et U([0, 1]).

19



1.0
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0.4
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−2

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0.2
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−4

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0.0
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−3 −2 −1 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0

Figure 7: Le modèle Gamma, N (0, 1) et U([0, 1]).

20



1.0
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q q q qqqqqqqqqqq
q q
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q q q qqqq q qqqqqqq q qq
q q qq
4

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q qq qq q q q q qqq qq q qq
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q
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q
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0.8
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2

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0.6
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0

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

Figure 8: Le modèle α-stable, N (0, 1) et U([0, 1]).

21



1.0
q q q q qq q qqq q q q
qq q qq
q q q q q qq q q qq qqq q qqq q qqq qq qqqq
qq q q q q q qq qqq
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3

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0.8
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2

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1

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q q q q qqqqqqqqqqqqqq q qqq q q qqq qq q
q q qq q qq
q qqq q q q qqqq q q q
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q qq qq q qqqqqq qqqq qq q q q qqq q
q q q qq qqqq qqqqqq qqqqqqqqqqqqqq q qq q qq
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0.4
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q qqq qqq q qq qqqqq qq q q
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q q q q q q q qq q
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0.0
q q
qq qq q
q q
q
q
q q q qq q q q qq q q q qqq
q q

−3 −2 −1 0 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0

Figure 9: Le modèle min-exponentielles, N (0, 1) et U([0, 1]).

22


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

23


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 .

24


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 10: Value of the copula on the border of the unit square.

25


une courte histoire des copules, et de la dépendance

Hoeffding, W. Schriften des
(1940). Masstabinvariante Korrelationstheorie.

Matematischen Instituts für Angewandte Matematik der Universitat Berlin. 5,
181-233.

26




181-233.

27


Fonction de répartition à marges uniformes

Z

Y
X

Figure 11: Graphical representation of a copula.

28


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

∂ 2 C(u, v)
c(u, v) = .
∂u∂v

29


Densité d’une loi à marges uniformes

z

x
x

Figure 12: Density of a copula.

30




181-233.

31


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


Figure 13: Distribution functions and densities.

32




Figure 14: Distribution functions and densities.

33


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.

34


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 15). And nally C
has uniform margins, and thus

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

35


Increasing functions in dimension 3

Figure 15: The notion of 3-increasing functions.

36


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

37


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.

38


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

39


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]).

40


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]).

41


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.

42


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 17: Contercomontonce, independent, and comonotone copulas.

43


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.

44


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 18: Mixing independence and comonotonicity.

45



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.

46


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



47



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.

48


If


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

comonotonic.

49


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



50


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 )

51


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.

52


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

53


Copulas in nance: options on multiple assets
Remark 11. Recall that Breeden Litzenberger (1978) proved that the risk
neutral probability can be obtrained from option prices: consider the price of a call
∞
Π(T, K) = e−rT EQ ((ST − K)+ ). Since (ST − K)+ = K 1(ST x)dx, one gets
∞
−rT
Π(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

54


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

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

55


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
Π(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
Π(T, K) = e e (T, x−K)−C e (T, x), e (T, x − K) dx.
−∞ ∂K ∂K ∂K

56


Natural properties for dependence measures
Denition 12. 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 → ∞.

57


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)).

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

58


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 - - - - - - - -

59


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 - - - - - - -

60


Les copules, un gadget ?
copulas: tales and facts, Mikosch, T. (2006)

61



62



63


1.0

4
0.8
Copule Gaussienne

2
0.6
0.4

0
0.2

!2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !2 0 2 4

Marges uniformes Marges gaussiennes

Figure 21: Simulations of the Gaussian copula (θ = 0.95).

64


1.0

4
0.8

2
Copule de Gumbel

0.6
0.4

0
0.2

!2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !2 0 2 4

Marges uniformes Marges gaussiennes

Figure 22: Simulations of Gumbel's copula θ = 1.2.

65



66



67



68



69



70



71


Risk measures and diversication
Any copula C is bounded by Fréchet-Hoeding bounds,

d
max ui − (d − 1), 0 ≤ C(u1 , . . . , ud ) ≤ min{u1 , . . . , ud },
i=1

and thus, any distribution F on F(F1 , . . . , Fd ) is bounded

d
max Fi (xi ) − (d − 1), 0 ≤ F (x1 , . . . , xd ) ≤ min{F1 (x1 ), . . . , Ff (xd )}.
i=1

Does this means the comonotonicity is always the worst-case scenario ?

Given a random pair (X, Y ), let (X − , Y − ) and (X + , Y + ) denote
contercomonotonic and comonotonic versions of (X, Y ), do we have

− −
? ?
R(φ(X , Y )) ≤ R(φ(X Y ) ≤ R(φ(X + , Y + )).
, )

72


Tchen's theorem and bounding some pure premiums
If φ : R2 → R is supermodular, i.e.

φ(x2 , y2 ) − φ(x1 , y2 ) − φ(x2 , y1 ) + φ(x1 , y1 ) ≥ 0,

for any x1 ≤ x2 and y1 ≤ y2 , then if (X, Y ) ∈ F(FX , FY ),

E φ(X − , Y − ) ≤ E (φ(X, Y )) ≤ E φ(X + , Y + ) ,

as proved in Tchen (1981).

Example 13. the stop loss premium for the sum of two risks E((X + Y − d)+ ) is
supermodular.

73


Example 14. For the n-year joint-life annuity,
n n
axy:n = v k P(Tx k and Ty k) = v k k pxy .
k=1 k=1

Then
a− +
xy:n ≤ axy:n ≤ axy:n ,

where
n
a−
xy:n = v k max{k px + k py − 1, 0}( lower Fréchet bound ),
k=1

n
a+
xy:n = v k min{k px , k py }( upper Fréchet bound ).
k=1

74


Example 15. For the n-year last-survivor annuity,
n n
axy:n = v k P(Tx k or Ty k) = v k k pxy ,
k=1 k=1

where k pxy = P(Tx k or Ty k) = k px + k py − k pxy .
Then
a−
xy:n
+
≤ axy:n ≤ axy:n ,
where
n
a−
xy:n = v k (1 − min{k qx , k qy }) ( upper Fréchet bound ),
k=1
n
a+
xy:n = v k (1 − max{k qx + k qy − 1, 0}) ( lower Fréchet bound ).
k=1

75


Example 16. For the widow's pension annuity,
∞ ∞
ax|y = ay − axy = v k k py − v k k pxy .
k=1 k=1

Then
a− ≤ ax|y ≤ a+ ,
x|y x|y

where
∞ ∞
a− = ay − axy =
x|y v k k py − v k min{k px , k py }.( upper Fréchet bound ),
k=1 k=1

∞ ∞
a+ = ay −axy =
x|y v k k py − v k max{k px + k py −1, 0}.( lower Fréchet bound ).
k=1 k=1

76


Value-at-Risk fails to be subadditive
One of the axiom of coherence is subadditivity, i.e. R(X + Y ) ≤ R(X) + R(Y ).
But Value-at-Risk is not coherent: one can have

VaR(X + Y, α)≥VaR(X, α) + VaR(Y, α) e.g. (see Embrechts (2007))

• X and Y are independent but very skew (see e.g. credit risk, with default

probabilities α),
• X and Y are independent but very heavy-tailed (see e.g. the case of innite

variance i.e. X ∈ Lp with p 1),
• X and Y are N (0, 1) with (special) tail dependence.

77


Makarov's theorem and bounding Value-at-Risk
In the case where R denotes the Value-at-Risk (i.e. quantile function of the PL

distribution),

R− ≤ R(X − + Y − )≤R(X + Y )≤R(X + + Y + ) ≤ R+ ,
where e.g. R+ can exceed the comonotonic case. Recall that

−1
R(X + Y ) = VaRq [X + Y ] = FX+Y (q) = inf{x ∈ R|FX+Y (x) ≥ q}.

If X ∼ E(α) and Y ∼ E(β),
P(X x) = exp(−x/α), P(Y y) = exp(−y/β) and x ∈ R+ .
The inequalities

exp(−x/ max{α, α}) ≤ Pr[X + Y x] ≤ exp(−(x − ξ)+ /(α + β))
hold for all x ∈ R+ , whatever the dependence between X and Y, where

ξ = (α + β) log(α + β) − α log α − β log β.

78


Therefore, the inequalities

− max{α, β} log(1 − q) ≤ VaRq [X + Y ] ≤ ξ − (α + β) log(1 − q)

hold for all q ∈ (0, 1).
Recall that in the independence case independence X + Y ∼ G(2, 1) and under

perfect positive dependence X + Y ∼ E(2).

79


Bornes de la VaR d’un portefeuille

4
2
0
!2
!4

0.0 0.2 0.4 0.6 0.8 1.0

Somme de 2 risques Gaussiens

Figure 23: Value-at-Risk for 2 Gaussian risks N (0, 1).

80



6
5
4
3
2
1
0

0.90 0.92 0.94 0.96 0.98 1.00

Somme de 2 risques Gaussiens

Figure 24: Value-at-Risk for 2 Gaussian risks N (0, 1).

81



20
15
10
5
0

0.0 0.2 0.4 0.6 0.8 1.0

Somme de 2 risques Gamma

Figure 25: Value-at-Risk for 2 Gamma risks G(3, 1).

82



20
15
10
5
0

0.90 0.92 0.94 0.96 0.98 1.00

Somme de 2 risques Gamma

Figure 26: Value-at-Risk for 2 Gamma risks G(3, 1).

83


Bounding Value-at-Risk for a sum of 2 risks
In a general (and theoretical) context, Schweizer Sklar (1981), studied the

distribution of ψ(X, Y ) for some R2 → R function ψ, where (X, Y ) ∈ F(FX , FY ),
using the concept of supremal and inmal convolutions,

Fsup (FX , FY ) (z) = sup {C (FX (x) , FY (y)) , ψ (x, y) = z} (1)

Finf (FX , FY ) (z) = inf {C (FX (x) , FY (y)) , ψ (x, y) = z} (2)

Williamson (1991) and Embrechts, Hoing Juri (2002) proposed

numerical algorithm to calculate those bounds.

The idea is to observe that bounds for the distribution of the sum of X and Y is

given by the distribution of Smin and Smax , where

P(Smax s) = sup max{P(X x) + P(Y s − x) − 1, 0}
x∈R

and

P(Smin ≤ s) = inf min{P(X ≤ x) + P(Y ≤ s − x), 1}.
x∈R

84


Note that those bounds can also be written as follows,

Proposition 17. Let (X, Y ) ∈ F(FX , FY ) then for all s ∈ R,
τC − (FX , FY )(s) ≤ P(X + Y ≤ s) ≤ ρC − (FX , FY )(s),

where
τC (FX , FY )(s) = sup {C(FX (x), FY (y)), x + y = s}
x,y∈R

and, if C(u, v) = u + v − C(u, v),
˜

˜
ρC (FX , FY )(s) = inf {C(FX (x), FY (y)), x + y = s}.
x,y∈R

85


Density of the sum of two uniform random variables Quantile of the sum of two uniform random variables

20
0.8

1#
67.81/0439:.074!.1!;/=
0.6
Density

1)
0.4

1(
0.2

?8@4A48@48B4
Independence C0.21,8D31.7E0$
Clayton, tau=0.5 C0.21,8D31.7E02
Clayton, tau=0.2 F7G-40D31.7E0$
Gumbel, tau=0.5 F7G-40D31.7E02
Gumbel, tau=0.2 C,G,8,1,8/B/12
0.0

12
Comonotonicity

0.0 0.5 1.0 1.5 2.0 0#0 0#$ 0%0 0%$ 100

*+,-.-/0/12304540

Figure 27: Sum to two U([0, 1]) random variables.

86


0.7 Density of the sum of three uniform random variables Quantile of the sum of three uniform random variables

30
0.6
0.5

2$
56-nti/e 89-/6e!-t!:is=
0.4
Density

20
0.3
0.2

ndependence
Independence B/-1ton, t-6D0$
Clayton, tau=0.5 B/-1ton, t-6D02
Clayton, tau=0.2 E6m,e/, t-6D0$

1$
0.1

Gumbel, tau=0.5 E6m,e/, t-6D02
Gumbel, tau=0.2 Bomonotonicit1
Comonotonicity
0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0#0 0#$ 0%0 0%$ 100

)ro,-,i/it1 /e4e/

Figure 28: Sum to three U([0, 1]) random variables.

87


0.5 Density of the sum of five uniform random variables Quantile of the sum of five uniform random variables

$!
0.4

4$
5uantil3 (9alu3!at!:is)
0.3

4!
Density

0.2

'$
Independence
Clayton, tau=0.5 nd3p3nd3nc3
0.1

Clayton, tau=0.2 Bla1tonC tauD!$
Gumbel, tau=0.5 Bla1tonC tauD!2
Gumbel, tau=0.2 Fumb3lC tauD!$

'!
Comonotonicity Fumb3lC tauD!2
Bomonotonicit1
0.0

0 1 2 3 4 5 !#! !#$ !%! !%$ !!

)robabilit1 l343l

Figure 29: Sum to ve U([0, 1]) random variables.

88


To go further
http://perso.univ-rennes1.fr/arthur.charpentier/slides-edf-2.pdf

89

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