1. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Gestion des risques bancaires et nanciers
risques extrêmes
et risques corrélés
Arthur Charpentier
EdF, formation continue
arthur.charpentier@univ-rennes1.fr
1
2. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
-3 -1 1 3
X
-1
3
Y
(Xi,Yi)
0.2 0.5 0.8
U (rank of X)
0.4
0.9
V(rankofY)
Density of the copula
Isodensity curves of the density
(Ui,Vi)
2
3. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Modelling correlated risks
• A short introduction to copulas
• Quantifying dependence
• Statistical inference
• Agregation properties
3
4. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Modelling correlated risks
• A short introduction to copulas
• Quantifying dependence
• Statistical inference
• Agregation properties
4
5. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
6. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Modelling correlated risks
• A short introduction to copulas
• Quantifying dependence
• Statistical inference
• Agregation properties
6
7. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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,
P(u1 U ≤ u2, v1 V ≤ v2) ≥ 0,
thus
C(u2, v2) − C(u1, v2)
−C(u2, v1) + C(u1, v1) ≥ 0. 0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Copula, positive area
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8. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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.
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9. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Borders of the copula function
!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
!0.20.00.20.40.60.81.01.21.4
!0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Figure 1: Value of the copula on the border of the unit square.
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10. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
X
Y
Z
Fonction de répartition à marges uniformes
Figure 2: Graphical representation of a copula.
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11. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
If C is twice dierentiable, one can dene its density as
c(u, v) =
∂2
C(u, v)
∂u∂v
.
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12. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
x x
z
Densité d’une loi à marges uniformes
Figure 3: Density of a copula.
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13. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes
Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes
Figure 4: Distribution functions and densities.
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14. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes
Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes
Figure 5: Distribution functions and densities.
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15. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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(F−1
X (u), F−1
Y (v)) for all (u, v) ∈ [0, 1] × [0, 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.
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16. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
z
sign(z)C(z) ≥ 0,
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].
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17. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Increasing functions in dimension 3
Figure 6: The notion of 3-increasing functions.
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18. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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(F−1
1 (u1), . . . , F−1
d (ud)) for all (ui) ∈ [0, 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,
c(u1, . . . , ud) =
∂d
C(u1, . . . , ud)
∂u1 . . . ∂ud
.
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19. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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.
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20. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Composante 1
p
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Composante 2
p
0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
Composante 3
p
Figure 7: Scatterplot in dimension 3 including projections.
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21. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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,
Fi(xi) =
#{observations Xi,j's lower than xi}
#{observations }
=
1
n
n
j=1
1(Xi,j ≤ xi).
Note that
Fi(Xi,j0
) =
#{observations Xi,j's lower than Xi,j0
}
#{observations }
=
1
n
n
j=1
1(Xi,j ≤ Xi,j0
) =
Ri,j0
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.
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22. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
5.56.06.57.07.58.08.59.0
Scatterplot of (X,Y)
X (raw data)
Y(rawdata)
5 10 15 20
5101520
Scatterplot of the ranks of (X,Y)
Ranks of the Xi’s
RanksoftheYi’s
0.2 0.4 0.6 0.8 1.0
0.20.40.60.81.0
Scatterplot of the ranks of (X,Y), divided by n
Ranks of the Xi’s/n
RanksoftheYi’s/n
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot o+ ,-,/0, t1e copula!t3pe tran+orm o+ ,6,70
Ui=Ranks of the Xi’s/n+1
Vi=RanksoftheYi’s/n+1
Figure 8: Copulas, ranks and parametric inference, from (Xi, Yi) to (Ui, Vi).
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23. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
(X, Y ) will be denoted (X⊥
, Y ⊥
). It is a random vector such that X⊥ L
= X and
Y ⊥ L
= 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
(X, Y ) will be denoted (X+
, Y +
). It is a random vector such that X+ L
= X and
Y + L
= Y , with copula C+
.
(X, Y ) has copula C+
if and only if there exists a strictly increasing function h
such that Y = h(X), or equivalently (X, Y )
L
= (F−1
X (U), F−1
Y (U)) where U is
U([0, 1]).
23
24. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
(X, Y ) will be denoted (X−
, Y −
). It is a random vector such that X− L
= X and
Y − L
= Y , with copula C−
.
(X, Y ) has copula C−
if and only if there exists a strictly decreasing function h
such that Y = h(X), or equivalently (X, Y )
L
= (F−1
X (1 − U), F−1
Y (U)) where U is
U([0, 1]).
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25. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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.
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27. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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,
(3, 2, 3) with probability 1/4,
(3, 1, 3) with probability 1/4,
then X and Y are independent, and Y and Z are independent, but X = Z.
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28. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 1 2 3 4
01234
X and Y independent
Component X
ComponentY
0 1 2 3 4
01234
Y and Z independent
Component Y
ComponentZ
0 1 2 3 4
01234
X and Z comonotonic
Component X
ComponentZ
Figure 10: Mixing independence and comonotonicity.
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29. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Pitfalls on independence and comonotonicity
The following proposition is false,
Uncorrect Proposition 8. If X and Y are comonotonic, if Y and Z are
comonotonic, then X and Z are comonotonic.
If
(X, Y, Z) = (1, 1, 1) with probability 1/4,
(1, 2, 3) with probability 1/4,
(3, 2, 1) with probability 1/4,
(3, 3, 3) with probability 1/4,
then X and Y are comonotonic, and Y and Z are comonotonic, but X and Z are
independent.
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30. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 1 2 3 4
01234
X and Y comonotonic
Component X
ComponentY
0 1 2 3 4
01234
Y and Z comonotonic
Component Y
ComponentZ
0 1 2 3 4
01234
X and Z independent
Component X
ComponentZ
Figure 11: Mixing independence and comonotonicity.
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31. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Pitfalls on independence and comonotonicity
The following proposition is false,
Uncorrect Proposition 9. If X and Y are comonotonic, if Y and Z are
independent, then X and Z are independent.
If
(X, Y, Z) = (1, 1, 3) with probability 1/4,
(2, 1, 1) with probability 1/4,
(2, 3, 3) with probability 1/4,
(3, 3, 1) with probability 1/4,
then X and Y are comonotonic, and Y and Z are independent, but X and Z are
anticomonotonic.
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32. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
If
(X, Y, Z) = (1, 1, 1) with probability 1/4,
(2, 1, 3) with probability 1/4,
(2, 3, 1) with probability 1/4,
(3, 3, 3) with probability 1/4,
then X and Y are comonotonic, and Y and Z are independent, but X and Z are
comonotonic.
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33. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 1 2 3 4
01234
X and Y comonotonic
Component X
ComponentY
0 1 2 3 4
01234
Y and Z independent
Component Y
ComponentZ
0 1 2 3 4
01234
X and Z comonotonic
Component X
ComponentZ
Figure 12: Mixing independence and comonotonicity.
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34. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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),
C(u, v) =
1
2π
√
1 − α2
Φ−1
(u)
−∞
Φ−1
(v)
−∞
exp
−(x2
− 2αxy + y2
)
2(1 − α2)
dxdy.
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,
C(u, v) =
1
2π
√
1 − α2
t−1
ν (u)
−∞
t−1
ν (v)
−∞
1 +
x2
− 2αxy + y2
2(1 − α2)
−((ν+2)/2)
dxdy.
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35. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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))
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36. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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.
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37. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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, ∞).
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38. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 5 10 15
05101520
Conditional independence, two classes
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, two classes
Figure 13: Two classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
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39. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 5 10 15 20 25 30
010203040
Conditional independence, three classes
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, three classes
Figure 14: Three classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
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40. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 20 40 60 80 100
020406080100
Conditional independence, continuous risk factor
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, continuous risk factor
Figure 15: Continuous classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
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41. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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.
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42. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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,
K(t) = t −
φ (t)
φ(t)
= t − λ(t),
which can be inverted easily in
φ(t) = φ(t0) exp
1
t0
1
λ(t)
dt ,
for some 0 t0 1 and 0 ≤ u ≤ 1.
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44. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Some characterizations of Archimedean copula
• Frank copula is the only Archimedean such that (U, V )
L
= (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),
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45. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Extreme value copulas
• Extreme value copulas
C(u, v) = exp (log u + log v) A
log u
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,
C(u1, . . . , ud) = C(u
1/z
1 , . . . , u
1/z
d )z
.
Those copula are then called max-stable: dene the maximum componentwise of
a sample X1, . . . , Xn, i.e. Mi = max{Xi,1, . . . , Xi,n}.
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46. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The joint distribution of M is
P(M ≤ x) = C(F1(x1, . . . , Fd(xd))n
,
where C is the copula of the Xi'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
.
Thus, C(u
1/n
1 , . . . , u
1/n
d )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
+ ωα
+ 1)
(
1/α),
for all 0 ≤ ω ≤ 1 and α ≥ 1.
46
47. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.50.60.70.80.91.0
Pickands dependence function A
Figure 16: Shape of Gumbel's dependence function A(ω).
47
48. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
49. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
Figure 17: Using geometric transformation to generate new copulas.
49
50. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
Figure 18: Using geometric transformation to generate new copulas.
50
51. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
Figure 19: Using geometric transformation to generate new copulas.
51
52. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
Figure 20: Using geometric transformation to generate new copulas.
52
53. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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) =
R
Cθ(u, v)dΠ(θ)
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
C(u, v) =
α2
2
C−
(u, v) + (1 − α2
)C⊥
(u, v) +
α2
2
C+
(u, v), α ∈ [0, 1].
53
54. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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) = Cn
(u1/n
, v1/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
55. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
Ch(u, v) = −
1
α
log 1 +
(e−αu
− 1)(e−αv
− 1)
e−α − 1
,
which is Frank copula.
55
56. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
57. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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,
P (Random1 ∈ ]a1, b1] , ..., Randomk ∈ ]ak, bk]) =
k
i=1
(bi − ai) .
Thus, one can generate easily random vectors U = (U1, ..., Ud) with independent
component.
57
58. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
60. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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 ∂1C(·|u1),
u2 ← [∂1C(·|u1)]−1
(Random2),
• simulate Uk from the conditional distribution ∂1,...,k−1C(·|u1, ..., uk−1),
uk ← [∂1,...,k−1C(·|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
61. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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) = ∂2C(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 ∂2C(·|u),
u2 ← [∂1C(·|u1)]−1
(Random2),
i.e.
u2 ← [(Random2)−α/(1+α
− 1]u−α
1 + 1−1/α
.
61
62. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
Distribution of v given u=0.3
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.5
Distribution of v given u=0.5
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
Generation of Clayton’s copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.5
Distribution of v given u=0.8
Figure 22: Simulation of Clayton's copula.
62
65. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Distribution de -
q/
0.0 0.2 0.4 0.6 0.8 1.0
0200400
Distribution de V
q/
0.0 0.2 0.4 0.6 0.8 1.0
0200400
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
-niform margins
!2 0 2 4
!4!202 tandard =aussian margins
Figure 25: Simulation of the contercomonotone copula.
65
66. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
!istri'tion de -
./
010 012 014 014 015 110
0200400
!istri'tion de 7
./
010 012 014 014 015 110
0100300900
010 012 014 014 015 110
010014015
-ni:orm mr=ins
!4 !2 0 2 4
!4!2024 Stndrd ?'ssin mr=ins
Figure 26: Simulation of the Gaussian copula.
66
67. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
D#$%u$on +e U
.y
010 012 014 014 015 110
0200400
D#$%u$on +e V
.y
010 012 014 014 015 110
0200400
010 012 014 014 015 110
010014015
Unfo%m ma%;n#
!2 0 2 4
!2024 S$an+a%+ =au##an ma%;n#
Figure 27: Simulation of Clayton's copula.
67
68. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Distribution de U
qy
0.0 0.2 0.4 0.6 0.8 1.0
0200400
Distribution de V
qy
0.0 0.2 0.4 0.6 0.8 1.0
0200400
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
Uniform margins
!4 !2 0 2 4
!4!2024 Standard Gaussian margins
Figure 28: Simulation of Clayton's survival copula.
68
69. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Distribution de U
qy
0.0 0.2 0.4 0.6 0.8 1.0
0200400
Distribution de V
qy
0.0 0.2 0.4 0.6 0.8 1.0
0200400
0.0 0.2 0.4 0.6 0.8 1.0
0.00.40.8
Uniform margins
!4 !2 0 2
!4!2024 Standard Gaussian margins
Figure 29: Simulation of a copula mixture.
69
70. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
C(T, K) = e−rT
∞
K
Q(ST x)dx,
hence
Q(ST ≤ x) = −e−rT ∂C
∂K
(T, x), or Q(ST ≤ x) = −erT ∂P
∂K
(T, x)
where P denotes the price of a put option.
Consider an option on 2 assets, with payo h(S1
T , S2
T ). The price at time 0 is
e−rT
EQ(h(S1
T , S2
T )).
70
71. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Copulas in nance: call on maximum
Here the payo is h(S1
T , S2
T ) = (max{S1
T , S2
T } − K)+. The price is then
C(T, K) = e−rT
EQ((max{S1
T , S2
T } − K)+)
= e−rT
EQ
∞
K
1 − 1(max{S1
T , S2
T } ≤ x)dx
= e−rT
∞
K
1 − Q(max{S1
T , S2
T } ≤ x)
Q(S1
T ≤x,S2
T ≤x)
dx,
hence, if (S1
T , S2
T ) has copula C (under Q), then
C(T, K) = e−rT
∞
K
1 − C erT ∂P1
∂K
(T, x), erT ∂P2
∂K
(T, x) dx.
71
72. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Copulas in nance: call on spreads
Here the payo is h(S1
T , S2
T ) = ([S1
T − S2
T ] − K)+. The price is then
C(T, K) = e−rT
EQ((S1
T − S2
T − K)+) = e−rT
EQ
∞
−∞
1(S2
T + K ≤ x ≤ S1
T )dx
= e−rT
∞
−∞
Q(K + S2
T ≤ x) − Q(S2
T + K ≤ x, S1
T ≤ x} ≤ x)
Q(S1
T ≤x,S2
T ≤x+K)
dx,
hence, if (S1
T , S2
T ) has copula C (under Q), then
C(T, K) = e−rT
∞
−∞
erT ∂P2
∂K
(T, x−K)−C erT ∂P1
∂K
(T, x), erT ∂P2
∂K
(T, x − K) dx.
72
73. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
74. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Modelling correlated risks
• A short introduction to copulas
• Quantifying dependence
• Statistical inference
• Agregation properties
74
75. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
76. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
As pointed out in Scarsini (1984), most of the axioms are self-evident.
If κ 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
77. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Association measures: Kendall's τ and Spearman's ρ
Rank correlations can be considered, i.e. Spearman's ρ dened as
ρ(X, Y ) = corr(FX(X), FY (Y )) = 12
1
0
1
0
C(u, v)dudv − 3
and Kendall's τ dened as
τ(X, Y ) = 4
1
0
1
0
C(u, v)dC(u, v) − 1.
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
78. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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)).
Equivalently, τ(X, Y ) = 1 −
4Q
n(n2 − 1)
where Q is the number of inversions
between the rankings of X and Y (number of discordance).
78
79. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
!2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0
!0.50.00.51.01.5 Concordant pairs
X
Y
!2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0
!0.50.00.51.01.5
Discordant pairs
X
Y
Figure 30: Concordance versus discordance.
79
80. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The case of the Gaussian random vector
If (X, Y ) is a Gaussian random vector with correlation r, then (Kruskal (1958))
ρ(X, Y ) =
6
π
arcsin
r
2
and τ(X, Y ) =
2
π
arcsin (r) .
80
81. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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
2
≤ ρ ≤
1 + 2τ − τ2
2
if τ ≥ 0
τ2
+ 2τ − 1
2
≤ ρ ≤
1 + 3τ
2
if τ ≤ 0.
which yield the area given below.
81
82. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
-1.0 -0.5 0.0 0.5 1.0
Tau de Kendall
-1.0
-0.5
0.0
0.5
1.0
RhodeSpearman
Figure 31: Admissible region of ρ and τ.
82
85. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Alternative expressions of those coecients
Note that those coecients can also be expressed as follows
ρ(X, Y ) =
[0,1]×[0,1]
C(u, v) − C⊥
(u, v)dudv
[0,1]×[0,1]
C+(u, v) − C⊥(u, v)dudv
(1)
(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)),
ρ(X) =
[0,1]d C(u) − C⊥
(u)du
[0,1]×[0,1]
C+(u) − C⊥(u)du
=
d + 1
2d − (d + 1)
2d
[0,1]d
C(u)du − 1
(2)
and similarly
τ(X) ==
1
2d−1 − 1)
2d
[0,1]d
C(u)dCu − 1 (3)
85
86. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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,
τ(X, Y, Z) =
1
3
(τ(X, Y ) + τ(X, Z) + τ(Y, Z)).
86
87. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
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 = Pr(U z|V z) = Pr(V z|U z),
and for upper tails,
R(z) = P(U z, V z)/(1 − z) = Pr(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
88. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Functional correlation measures
Consider also Kendall's tau, dened as −1 + 4
1
0
1
0
C(u, v)dC(u, v).
Denition 25. The cumulative tau can be dened as
J(z) = −1 + 4
z
0
z
0
C(u, v)dC(u, v)/C(z, z)2
.
88
94. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Dependence in independence
Coles, Heffernan Tawn (1999) propose another function,
χ(z) =
2 log(1 − z)
log C(z, z)
− 1
Then set η = (1 + limz→1 χ(z))/2
94
99. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
(Strong) tail dependence measure
Joe (1993) dened, in the bivariate case a tail dependence measure.
Denition 26. Let (X, Y ) denote a random pair, the upper and lower tail
dependence parameters are dened, if the limit exist, as
λL = lim
u→0
P X ≤ F−1
X (u) |Y ≤ F−1
Y (u) ,
and
λU = lim
u→1
P X F−1
X (u) |Y F−1
Y (u) .
As mentioned in Fougères (2004), this coecient can be obtained dierently:
set
θ(x) =
log P(max{X, Y } ≤ x)
log P(X ≤ x)
.
Then
λU = 2 − lim
x→∞
θ(x),
99
100. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
since when x → ∞
2 −
log P(max{X, Y } ≤ x)
log P(X ≤ x)
∼
P(X x, Y x)
1 − P(X x)
= P(Y x|X x).
Note that these coecient can be expressed only through the copula,
Proposition 27. Let (X, Y ) denote a random pair with copula C, the upper and
lower tail dependence parameters are dened, if the limit exist, as
λL = lim
u→0
C(u, u)
u
and λU = lim
u→1
C∗
(u, u)
1 − u
.
Does λ = 0 implies that extremal events are independent ?
Example 28. If (X, Y ) has a Gaussian copula with parameter θ 1, then λ = 0.
Hence, visually, dependence is weaker than any Gumbel's copula (even with θ is
rather small), but are extremal events independent ?
100
101. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Marges uniformes
CopuledeGumbel
!2 0 2 4
!2024
Marges gaussiennes
Figure 41: Simulations of Gumbel's copula θ = 1.2.
101
102. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Marges uniformes
CopuleGaussienne
!2 0 2 4
!2024
Marges gaussiennes
Figure 42: Simulations of the Gaussian copula (θ = 0.95).
102
103. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Example 29. Consider the case of Archimedean copulas, then
λU = 2 − lim
x→0
1 − φ−1
(2x)
1 − φ−1(x)
and λL = lim
x→0
φ−1
(2φ(x))
x
= lim
x→∞
φ−1
(2x)
φ−1(x)
.
Further, properties can be derived for distorted generators, φα,β(·) = φ(·α
)β
,
upper and lower tails coecients are respectively
λU and λ
1/α
L for φα,1(·) = φ(·α
)
and
2 − (2 − λU )1/β
and λ
1/β
L for φ1,β(·) = φ(·)β
(Weak) tail dependence measure
Ledford Tawn (1996) propose the following model to study tail dependence.
Consider a random vector with identically distributed marginals, X
L
= Y .
• under independence, P(X t, Y t) = P(X t) × P(Y t) = P(X t)2
,
• under comonotonicity, P(X t, Y t) = P(X t) = P(X t)1
,
103
104. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Assume that P(X t, Y t) ∼ P(X t)1/η
as t → ∞, where η ∈ (0, 1] will be
called coecient of tail dependence
More precisely,
• η = 1, perfect positive dependence (tail comontonicity),
• 1/2 η 1, more dependent than independence, but asymptotically
independent,
• η = 1/2, tail independence
• 0 η 1/2 less dependent than independence.
104
105. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
One can then dene upper tail coecient ηU and lower tail coecient ηL.
Example 30. : If (X, Y ) has Gumbel copula,
P(X ≤ x, Y ≤ y) = exp(−(x−α
+ y−α
)1/α
), α ≥ 0
then ηU = 1. Further, ηL = 1/2α
.
Example 31. : If (X, Y ) has a Clayton copula, then ηU = 1/2 while ηL = 1.
Example 32. : If (X, Y ) has a Gaussian copula, then ηU = ηL = (1 + r)/2.
105
106. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Modelling correlated risks
• A short introduction to copulas
• Quantifying dependence
• Statistical inference
• Agregation properties
106
107. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Estimation of copulas
Since C(u, v) = F(F−1
X (u), F−1
Y (v)), copula has be estimated only after
estimating marginal distribution.
Margins Copula
Parametric Fα and Fβ C
θ
Nonparametric FX and FY C
(Fully) parametric estimation of copulas
Step 1: t the 2 univariate marginal cdf's FX and FY with the help of the
observations {x1, x2, . . . , xn} and {y1, y2, . . . , yn} respectively; let α and β be the
corresponding MLE's of α and β.
Step 2: estimate θ with the parameters α = α and β = β xed at the estimated
values from Step 1; i.e. on pseudo-observations (Ui, Vi)'s, where
Ui = Fα(Xi) and Vi = Fβ(Yi),
107
108. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
let the result be θ.
Step 3: using α, β and θ as starting values, determine the global MLE's α, β and
θ of the parameters α, β and θ.
Parametric estimation of copulas
An alternative is to use nonparametric estimation of margins, FX and FY .
Step 1: estimate θ based on pseudo-observations (Ui, Vi)'s, where
Ui = FX(Xi) and Vi = FY (Yi),
let the result be θ.
Nonparametric estimation of copulas
Given an estimation of marginal distributions (parametric or nonparametric), the
108
109. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
idea is to consider the empirical copula C, dened as
C(u, v) =
#{i such that Ui ≤ u and Vi ≤ v}
#{i}
=
1
n
n
i=1
1(FX(Xi) ≤ u) × 1(FY (Yi) ≤ v).
109
110. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Example Loss-ALAE: consider the following dataset, were the Xi's are loss
amount (paid to the insured) and the Yi's are allocated expenses. Denote by Ri
and Si the respective ranks of Xi and Yi. Set Ui = Ri/n = ˆFX(Xi) and
Vi = Si/n = ˆFY (Yi).
Figure 43 shows the log-log scatterplot (log Xi, log Yi), and the associate copula
based scatterplot (Ui, Vi).
Figure 44 is simply an histogram of the (Ui, Vi), which is a nonparametric
estimation of the copula density.
Note that the histogram suggests strong dependence in upper tails (the
interesting part in an insurance/reinsurance context).
110
111. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
1 2 3 4 5 6
12345 Log!log scatterplot, Loss!ALAE
log(LOSS)
log(ALAE)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Copula type scatterplot, Loss!ALAE
Probability level LOSS
ProbabilitylevelALAE
Figure 43: Loss-ALAE, scatterplots (log-log and copula type).
111
112. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Figure 44: Loss-ALAE, histogram of copula type transformation.
112
113. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The basic idea to get an estimator of the density at some point x is to count how
many observation are in the neighborhood of x (e.g. in [x − h, x + h) for some
h 0).
Therefore, consider the moving histogram or naive estimator as suggested by
Rosenblatt (1956),
f(x) =
1
2nh
n
i=1
I(Xi ∈ [x − h, x + h)).
Note that this can be easily extended using other denitions of the neighborhood
of x,
f(x) =
1
nh
n
i=1
K
x − Xi
h
,
where K is a kernel function (e.g. K(ω) = I(|ω| ≤ 1)/2).
113
114. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0
1
2
3
4
5
Estimation of Frank copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Figure 45: Theoretical density of Frank copula.
114
115. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0
1
2
3
4
5
Estimation of Frank copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Figure 46: Estimated density of Frank copula, using standard Gaussian (indepen-
dent) kernels, h = h∗
.
115
116. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Problem of nonparametric estimation with kernel: bias on the borders.
Let K denote a symmetric kernel with support [−1, 1]. Note that
E(f(0, h) =
1
2
f(0) + O(h)
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117. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.01.2 Kernel based estimation of the uniform density on [0,1]
Density
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.01.2
Kernel based estimation of the uniform density on [0,1]
Density
Figure 47: Density estimation of an uniform density on [0, 1].
117
118. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Several techniques have been introduce to get a better estimation on the border,
• boundary kernel (Müller (1991))
• mirror image modication (Deheuvels Hominal (1989), Schuster
(1985))
• transformed kernel (Devroye Györfi (1981), Wand, Marron
Ruppert (1991))
• Beta kernel (Brown Chen (1999), Chen (1999, 2000)),
see Charpentier, Fermanian Scaillet (2006) for a survey with
application on copulas.
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119. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Consider the kernel estimator of the density of the
(Xi, Yi) = (G−1
(Ui), G−1
(Vi))'s, where G is a strictly increasing distribution
function R → [0, 1], with a dierentiable density.
119
120. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Since density f of (X, Y ) is continuous, twice dierentiable, and bounded above,
for all (x, y) ∈ R2
, consider
f(x, y) =
1
nh2
n
i=1
K
x − Xi
h
K
y − Yi
h
.
Since
f(x, y) = g(x)g(y)c[G(x), G(y)]. (4)
can be inverted in
c(u, v) =
f(G−1
(u), G−1
(v))
g(G−1(u))g(G−1(v))
, (u, v) ∈ [0, 1] × [0, 1], (5)
one gets, substituting f in (5)
c(u, v) =
1
nh · g(G−1(u)) · g(G−1(v))
n
i=1
K
G−1
(u) − G−1
(Ui)
h
,
G−1
(v) − G−1
(Vi)
h
,
(6)
120
121. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0
1
2
3
4
5
Estimation of Frank copula
0.2 0.4 0.6 0.8
0.20.40.60.8
Figure 48: Estimated density of Frank copula, using a Gaussian kernel, after a
Gaussian normalization.
121
122. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The Beta-kernel based estimator of the copula density at point (u, v), is obtained
using product beta kernels, which yields
c(u, v) =
1
n
n
i=1
K Xi,
u
b
+ 1,
1 − u
b
+ 1 · K Yi,
v
b
+ 1,
1 − v
b
+ 1 ,
where K(·, α, β) denotes the density of the Beta distribution with parameters α
and β.
122
124. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Estimation of the copula density (Beta kernel, b=0.1) Estimation of the copula density (Beta kernel, b=0.1)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Figure 50: Estimated density of Frank copula, Beta kernels, b = 0.1
124
125. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Estimation of the copula density (Beta kernel, b=0.05) Estimation of the copula density (Beta kernel, b=0.05)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Figure 51: Estimated density of Frank copula, Beta kernels, b = 0.05
125
126. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
01234
Standard Gaussian kernel estimator, n=100
Estimation of the density on the diagonal
Densityoftheestimator
0.0 0.2 0.4 0.6 0.8 1.0
01234
Standard Gaussian kernel estimator, n=1000
Estimation of the density on the diagonal
Densityoftheestimator
0.0 0.2 0.4 0.6 0.8 1.0
01234
Standard Gaussian kernel estimator, n=10000
Estimation of the density on the diagonal
Densityoftheestimator
Figure 52: Density estimation on the diagonal, standard kernel.
126
127. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
01234
Transformed kernel estimator (Gaussian), n=100
Estimation of the density on the diagonal
Densityoftheestimator
0.0 0.2 0.4 0.6 0.8 1.0
01234
Transformed kernel estimator (Gaussian), n=1000
Estimation of the density on the diagonal
Densityoftheestimator
0.0 0.2 0.4 0.6 0.8 1.0
01234
Transformed kernel estimator (Gaussian), n=10000
Estimation of the density on the diagonal
Densityoftheestimator
Figure 53: Density estimation on the diagonal, transformed kernel.
127
128. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0.0 0.2 0.4 0.6 0.8 1.0
01234
Beta kernel estimator, b=0.05, n=100
Estimation of the density on the diagonal
Densityoftheestimator
0.0 0.2 0.4 0.6 0.8 1.0
01234
Beta kernel estimator, b=0.02, n=1000
Estimation of the density on the diagonal
Densityoftheestimator
0.0 0.2 0.4 0.6 0.8 1.0
01234
Beta kernel estimator, b=0.005, n=10000
Estimation of the density on the diagonal
Densityoftheestimator
Figure 54: Density estimation on the diagonal, Beta kernel.
128
129. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Tail dependence and statistical inference
Consider an i.i.d. sample (X1, Y1) , ...., (Xn, Yn).
Consider unit Pareto transformation of margins: set
T =
1
1 − FX (X)
∧
1
1 − FY (Y )
.
Observe that the survival distribution function of T, FT , is regularly varying
with parameter η. But because FX and FY are unknown, dene the pseudo
observations Ti's as
Ti =
1
1 − FX,n (Xi)
∧
1
1 − FY,n (Yi)
=
n + 1
n + 1 − Ri
∧
n + 1
n + 1 − Si
,
where Ri and Si denote the ranks of the Xi's and Yi's. Hill estimator can then
be used, based on the k + 1 largest values of the Ti's,
ηHill =
1
k
k
i=1
log
Tn−i+1:n
Tn−k:n
.
129
130. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Estimation of η (Hill's estimator)
Proposition 33. Assume that (X, Y ) has upper tail dependence, with tail index
η, with additional regularity conditions, then
√
k (ηHill − η) is asymptotically
normally distributed, with mean 0 and variance
σ2
= η2
(1 − l) 1 − 2
∂c (1, 1)
∂x
∂c (1, 1)
∂y
.
Remark 34. From this Proposition, a test for asymptotic dependence (i.e.
η = 1) can de dened: asymptotic dependence is accepted if
1 − ηHill
σ (η = 1)
≤ Φ−1
(95%)
130
131. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Estimation of η (Peng's estimator)
Set
Sn (k) =
n
i=1
I (Xi Xn−k:n, Yi Yn−k:n)
and
ηPeng =
1
log 2
log
Sn (k)
Sn ( k/2 )
−1
.
Proposition 35. Assume that (X, Y ) has upper tail dependence, with tail index
η, and the same technical assumption as before, then
√
k (ηPeng − η) is
asymptotically normally distributed, with mean 0.
131
132. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Estimation of λ (Huang's estimator)
It is also possible to estimate λU : Huang-estimator of is based on the denition
of the upper tail index.
λHuang =
n
k
P (Ui, Vi) ∈ 1 −
k
n
× 1 −
k
n
=
1
k
n
i=1
I (Ri n − k, Si n − k)
=
1
k
n
i=1
I (Xi Xn−k:n, Yi Yn−k:n) .
132
133. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Hill estimate), Gaussian copula, tau=0.3
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Peng estimate), Gaussian copula, tau=0.3
0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
1.0
Lambda (Huang estimate), Gaussian copula, tau=0.3
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Hill estimate), Clayton copula, tau=0.3
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Peng estimate), Clayton copula, tau=0.3
0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
1.0
Lambda (Huang estimate), Clayton copula, tau=0.3
Figure 55: Estimation of η and λ for Gaussian and Clayton copulas, with Kendall's
tau equal to 0.3
133
134. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Hill estimate), survival Clayton copula, tau=0.3
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Peng estimate), survival Clayton copula, tau=0.3
0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
1.0
Lambda (Huang estimate), survival Clayton copula, tau=0.3
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Hill estimate), Gumbel copula, tau=0.3
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Peng estimate), Gumbel copula, tau=0.3
0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
1.0
Lambda (Huang estimate), Gumbel copula, tau=0.3
Figure 56: Estimation of η and λ for survival Clayton and Gumbel copulas, with
Kendall's tau equal to 0.3
134
135. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Hill estimate), Gaussian copula, tau=0.7
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Peng estimate), Gaussian copula, tau=0.7
0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
1.0
Lambda (Huang estimate), Gaussian copula, tau=0.7
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Hill estimate), Clayton copula, tau=0.7
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Peng estimate), Clayton copula, tau=0.7
0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
1.0
Lambda (Huang estimate), Clayton copula, tau=0.7
Figure 57: Estimation of η and λ for Gaussian and Clayton copulas, with Kendall's
tau equal to 0.7
135
136. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Hill estimate), survival Clayton copula, tau=0.7
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Peng estimate), survival Clayton copula, tau=0.7
0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
1.0
Lambda (Huang estimate), survival Clayton copula, tau=0.7
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Hill estimate), Gumbel copula, tau=0.7
0 100 200 300 400
0.5
0.7
0.9
1.1
Eta (Peng estimate), Gumbel copula, tau=0.7
0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
1.0
Lambda (Huang estimate), Gumbel copula, tau=0.7
Figure 58: Estimation of η and λ for survival Clayton and Gumbel copulas, with
Kendall's tau equal to 0.7
136
137. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Agenda
• General introduction
Modelling correlated risks
• A short introduction to copulas
• Quantifying dependence
• Statistical inference
• Agregation properties
137
138. Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Risk measures and diversication
Any copula C is bounded by Fréchet-Hoeding bounds,
max
d
i=1
ui − (d − 1), 0 ≤ C(u1, . . . , ud) ≤ min{u1, . . . , ud},
and thus, any distribution F on F(F1, . . . , Fd) is bounded
max
d
i=1
Fi(xi) − (d − 1), 0 ≤ F(x1, . . . , xd) ≤ min{F1(x1), . . . , Ff (xd)}.
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 +
)).
138