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- 1. Arthur CHARPENTIER - cole dt EURIA. mesures de risques et
dpendance Arthur Charpentier Universit de Rennes 1 & cole
Polytechnique arthur.charpentier@univ-rennes1.fr
http://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/
1
- 2. Arthur CHARPENTIER - cole dt EURIA. 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
- 3. Arthur CHARPENTIER - cole dt EURIA. Agenda General
introductionModelling correlated risks A short introduction to
copulas Quantifying dependence Statistical inference Agregation
properties 3
- 4. Arthur CHARPENTIER - cole dt EURIA. Agenda General
introductionModelling correlated risks A short introduction to
copulas Quantifying dependence Statistical inference Agregation
properties 4
- 5. Arthur CHARPENTIER - cole dt EURIA. Some references on large
and correlated risksRank , 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 inFinance.
Wiley,Beirlant , J., Goegebeur , Y., Segers, J. & Teugels , J.
(2004). Statistics ofExtremes: Theory and Applications.
Wiley,McNeil , A. Frey , R., & Embrechts , P. (2005).
Quantitative RiskManagement: Concepts, Techniques, and Tools.
Princeton University Press, 5
- 6. Arthur CHARPENTIER - cole dt EURIA. Agenda General
introductionModelling correlated risks A short introduction to
copulas Quantifying dependence Statistical inference Agregation
properties 6
- 7. Arthur CHARPENTIER - cole dt EURIA. Copulas, an introduction
(in dimension 2)Denition 1. A copula C is a joint distribution
function on [0, 1]2 , with uniformmargins on [0, 1].Set C(u, v) =
P(U u, V v), where (U, V ) is a random pair with uniformmargins.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 andv1 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
- 8. Arthur CHARPENTIER - cole dt EURIA.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) =
uand 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
- 9. Arthur CHARPENTIER - cole dt EURIA. 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
- 10. Arthur CHARPENTIER - cole dt EURIA. Fonction de rpartition
marges uniformes Z Y X Figure 2: Graphical representation of a
copula. 10
- 11. Arthur CHARPENTIER - cole dt EURIA.If C is twice
dierentiable, one can dene its density as 2 C(u, v) c(u, v) = . uv
11
- 12. Arthur CHARPENTIER - cole dt EURIA. Densit dune loi marges
uniformes z x x Figure 3: Density of a copula. 12
- 13. Arthur CHARPENTIER - cole dt EURIA. Fonction de rpartition
marges uniformes Densit dune loi marges uniformes Fonction de
rpartition marges uniformes Densit dune loi marges uniformes Figure
4: Distribution functions and densities. 13
- 14. Arthur CHARPENTIER - cole dt EURIA. Fonction de rpartition
marges uniformes Densit dune loi marges uniformes Fonction de
rpartition marges uniformes Densit dune loi marges uniformes Figure
5: Distribution functions and densities. 14
- 15. Arthur CHARPENTIER - cole dt EURIA. Sklars theoremTheorem
3. (Sklar) Let C be a copula, and FX and FY two
marginaldistributions, then F (x, y) = C(FX (x), FY (y)) is a
bivariate distributionfunction, with F F(FX , FY ).Conversely, if F
F(FX , FY ), there exists C such thatF (x, y) = C(FX (x), FY (y)).
Further, if FX and FY are continuous, then C isunique, and given by
C(u, v) = F (FX (u), FY (v)) for all (u, v) [0, 1] [0, 1] 1 1We
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 forany increasing functions g and h.
15
- 16. Arthur CHARPENTIER - cole dt EURIA. Copulas, an
introduction (in dimension d 2)Denition 5. A copula C is a joint
distribution function on [0, 1]d , withuniform 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 somei {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, zwhere the sum
is taken over all vertices of [u v], and where sign(z) is +1 ifzi =
ui for an even number of i (and 1 otherwise, see Figure 6). And
nally Chas uniform margins, and thus C(1, . . . , 1, ui , 1, . . .
, 1) = ui on [0, 1]. 16
- 17. Arthur CHARPENTIER - cole dt EURIA. Increasing functions in
dimension 3 Figure 6: The notion of 3-increasing functions. 17
- 18. Arthur CHARPENTIER - cole dt EURIA.Theorem 6. (Sklar) Let C
be a copula, and F1 , . . . , Fd be d marginaldistributions, then F
(x) = C(F1 (x1 ), . . . , Fd (xd )) is a distribution function,
withF F(F1 , . . . , Fd ).Conversely, if F F(F1 , . . . , Fd ),
there exists C such thatF (x) = C(F1 (x1 ), . . . , Fd (xd )).
Further, if the Fi s are continuous, then C isunique, and given by
C(u) = F (F1 (u1 ), . . . , Fd (ud )) for all (ui ) [0, 1] 1 1We
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
ofU = (F1 (X1 ), . . . , Fd (Yd )).Again, if C is dierentiable, one
can dene its density, d C(u1 , . . . , ud ) c(u1 , . . . , ud ) = .
u1 . . . ud 18
- 19. Arthur CHARPENTIER - cole dt EURIA. Copulas in high
dimension, a dicult problemIt is usually dicult to represent
dependence in dimension d > 2, and it isusually studied by
pairs.In dimension d = 2, one can dene the following Frchet class
F(FX , FY , FZ )dened by its marginal distributions. But it can
also be interested to studyF(FXY , FXZ , FY Z ) dened by it paired
distributions.One of the problem that arises is the compatibility
of marginals: one has toverify that CXY (x, y) = CX|Z (x|z)CY |Z
(y|z)dz,for instance. 19
- 20. Arthur CHARPENTIER - cole dt EURIA. 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
- 21. Arthur CHARPENTIER - cole dt EURIA. Copulas and ranksThe
copula of X = (X1 , . . . , Xd ) is the distribution function ofU =
(F1 (X1 ), . . . , Fd (Yd )).In practice, since marginal
distributions are unknown, the idea is to substituteempirical
distribution function, n #{observations Xi,j s lower than xi } 1 Fi
(xi ) = = 1(Xi,j xi ). #{observations } n j=1Note that n
#{observations Xi,j s lower than Xi,j0 } 1 Ri,j0Fi (Xi,j0 ) = =
1(Xi,j Xi,j0 ) = , #{observations } n j=1 nwhere 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
- 22. Arthur CHARPENTIER - cole dt EURIA. Scatterplot of (X,Y)
Scatterplot of the ranks of (X,Y) 9.0 20 8.5 8.0 15 Ranks of the
Yis 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 Xis 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 Yis/n+1 Ranks of
the Yis/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 Xis/n Ui=Ranks of the Xis/n+1Figure 8:
Copulas, ranks and parametric inference, from (Xi , Yi ) to (Ui ,
Vi ). 22
- 23. Arthur CHARPENTIER - cole dt EURIA. Some very classical
copulas The independent copula C(u, v) = uv = C (u, v).The copula
is standardly denoted , 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 LY =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 LY+ =Y, with
copula C +.(X, Y ) has copula C+ if and only if there exists a
strictly increasing function h L1 1such that Y = h(X), or
equivalently (X, Y ) = (FX (U ), FY (U )) where U isU([0, 1]).
23
- 24. Arthur CHARPENTIER - cole dt EURIA.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 ofcopulas.In higher dimension, C + (u1 , . . . ,
ud ) = min{u1 , . . . , ud } is the comonotoniccopula. 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 LY =Y, with copula C .(X, Y ) has copula C if
and only if there exists a strictly decreasing function h L1 1such
that Y = h(X), or equivalently (X, Y ) = (FX (1 U ), FY (U )) where
U isU([0, 1]). 24
- 25. Arthur CHARPENTIER - cole dt EURIA.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 sinceC(u, v) 0, C(u, v) max{u + v 1, 0} = C (u, v).
Thus, C is a lower boundfor the set of copulas.In higher dimension,
C (u1 , . . . , ud ) = max{u1 + . . . + ud (d 1), 0} is not
acopula: if (X, Y ) and (X, Z) are countercomonotonic, (Y, Z) is
necessarilycomonotonic - it is not possible to have all component
highly negativelycorrelated.Anyway, it is still the best pointwise
lower bound. 25
- 26. Arthur CHARPENTIER - cole dt EURIA. 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
bou0 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 Frchet Lower Bound Independent copula
Frchet Upper Bound1.0 1.0 1.00.8 0.8 0.80.6 0.6 0.60.4 0.4 0.40.2
0.2 0.20.0 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 Frchet!Hoeffding bound
Scatterplot, Indepedent copula random generation Scatterplot, Upper
Frchet!Hoeffding bound1.0 1.0 1.00.8 0.8 0.80.6 0.6 0.60.4 0.4
0.40.2 0.2 0.20.0 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
- 27. Arthur CHARPENTIER - cole dt EURIA. Pitfalls on
independence and comonotonicityThe following proposition is
false,Uncorrect Proposition 7. If X and Y are independent, if Y and
Z areindependent, 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. 27
- 28. Arthur CHARPENTIER - cole dt EURIA. X and Y independent Y
and Z independent X and Z comonotonic 4 4 4 3 3 3Component 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
- 29. Arthur CHARPENTIER - cole dt EURIA. Pitfalls on
independence and comonotonicityThe following proposition is
false,Uncorrect Proposition 8. If X and Y are comonotonic, if Y and
Z arecomonotonic, 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
areindependent. 29
- 30. Arthur CHARPENTIER - cole dt EURIA. X and Y comonotonic Y
and Z comonotonic X and Z independent 4 4 4 3 3 3Component 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 11: Mixing
independence and comonotonicity. 30
- 31. Arthur CHARPENTIER - cole dt EURIA. Pitfalls on
independence and comonotonicityThe following proposition is
false,Uncorrect Proposition 9. If X and Y are comonotonic, if Y and
Z areindependent, 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
areanticomonotonic. 31
- 32. Arthur CHARPENTIER - cole dt EURIA.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
arecomonotonic. 32
- 33. Arthur CHARPENTIER - cole dt EURIA. X and Y comonotonic Y
and Z independent X and Z comonotonic 4 4 4 3 3 3Component 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 12: Mixing
independence and comonotonicity. 33
- 34. Arthur CHARPENTIER - cole dt EURIA. Elliptical (Gaussian
and t) copulasThe idea is to extend the multivariate probit model,
Y = (Y1 , . . . , Yd ) withmarginal B(pi ) distributions, modeled
as Yi = 1(Xi ui ), where X N (I, ). The Gaussian copula, with
parameter (1, 1), 1 (u) 1 (v) 1 (x2 2xy + y 2 ) C(u, v) = exp dxdy.
2 1 2 2(1 2 )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 (1, 1) and 2,
t1 (u) t1 (v) ((+2)/2) 1 x2 2xy + y 2 C(u, v) = 1+ dxdy. 2 1 2 2(1
2 ) 34
- 35. Arthur CHARPENTIER - cole dt EURIA. Archimedean copulas
Denition of Archimedean copulas Archimedian copulas C(u, v) = 1
((u) + (v)), where is decreasing convex (0, 1), with (0) = and (1)
= 0.Example 10. If (t) = [ log t] , then C is Gumbels copula, and
if(t) = t 1, C is Claytons. Note that C is obtained when (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 anheterogeneous component . Assume further that
P(X x| = ) = (GX (x)) and P(Y y| = ) = (GY (y))for some baseline
distribution functions GX and GY .Then F (x, y) = P(X x, Y y) =
E(P(X x, Y y| = )) 35
- 36. Arthur CHARPENTIER - cole dt EURIA.thus, since X and Y are
conditionally independent, F (x, y) = E(P(X x| = ) P(Y y| = ))and
therefore F (x, y) = E (GX (x)) (GY (y)) = ( log GX (x) log GY
(y))where denotes the Laplace transform of , i.e. (t) = E(et ).
Since FX (x) = ( log GX (x)) and FY (y) = ( log GY (y))and thus,
the joint distribution of (X, Y ) satises F (x, y) = ( 1 (FX (x)) +
1 (FY (y))).Example 11. If is Gamma distributed, the associated
copula is Claytons. If has a stable distribution, the associated
copula is Gumbels. 36
- 37. Arthur CHARPENTIER - cole dt EURIA.Consider two risks, X
and Y, such that X| = G E(G ) and Y | = G E(G ) are independent, X|
= B E(B ) and Y | = B E(B ) are independent,(unobservable good (G)
and bad (B ) risks).The following gures start from 2 classes of
risks, then 3, and then a continuousrisk factor (0, ). 37
- 38. Arthur CHARPENTIER - cole dt EURIA. 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 (1 (FX (Xi )), 1 (FY (Yi ))).
38
- 39. Arthur CHARPENTIER - cole dt EURIA. 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
3Figure 14: Three classes of risks, (Xi , Yi ) and (1 (FX (Xi )), 1
(FY (Yi ))). 39
- 40. Arthur CHARPENTIER - cole dt EURIA. 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 3Figure 15: Continuous classes of risks,
(Xi , Yi ) and (1 (FX (Xi )), 1 (FY (Yi ))). 40
- 41. Arthur CHARPENTIER - cole dt EURIA.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
isClaytons. If S is the Weibull survival distribution, the
associated copula isGumbels. 41
- 42. Arthur CHARPENTIER - cole dt EURIA.3. The use of Kendalls
distribution function K(t) = P(C(U, V ) t) where(U, V ) is a random
pair with distribution function C .Then, for Archimedean copulas,
(t) K(t) = t = t (t), (t)which can be inverted easily in 1 1 (t) =
(t0 ) exp dt , t0 (t)for some 0 < t0 < 1 and 0 u 1. 42
- 43. Arthur CHARPENTIER - cole dt EURIA. Some more examples of
Archimedean copulas (t) range (1) 1 (t 1) [1, 0) (0, ) Clayton,
Clayton (1978) (2) (1 t) [1, ) 1(1t) (3) log [1, 1) Ali-Mikhail-Haq
Gumbel Hougaard t (4) ( log t) [1, ) Gumbel, (1960), (1986) (5) log
e t 1 e 1 (, 0) (0, ) Frank, Frank (1979), Nelsen(1987) (6) log{1
(1 t) } [1, ) Joe, Frank (1981), Joe(1993) (7) log{t + (1 )} (0, 1]
(8) 1t [1, ) 1+(1)t (9) log(1 log t) (0, 1] Barnett (1980), Gumbel
(1960) (10) log(2t 1) (0, 1] (11) log(2 t ) (0, 1/2] (12) ( 1 1)
[1, ) t (13) (1 log t) 1 (0, ) (14) (t1/ 1) [1, ) (15) (1 t1/ ) [1,
) Genest & Ghoudi (1994) (16) ( + 1)(1 t) [0, ) t 43
- 44. Arthur CHARPENTIER - cole dt EURIA. 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
- 45. Arthur CHARPENTIER - cole dt EURIA. 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 , } A () 1 for all [0, 1] .An alternative
denition is the following: C is an extreme value copula if for allz
> 0, 1/z 1/z C(u1 , . . . , ud ) = C(u1 , . . . , ud )z .Those
copula are then called max-stable: dene the maximum componentwise
ofa sample X 1 , . . . , Xn , i.e. Mi = max{Xi,1 , . . . , Xi,n }.
45
- 46. Arthur CHARPENTIER - cole dt EURIA.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/nThus, C(u1 , . . . , ud )n is the copula of the n maximum
componentwise from asample with copula C .Example 13. : If A is
constant (1 on [0, 1]), then X and Y are independent,and if A() =
max {, 1 }, X and Y are comonotonic. Gumbels copula isobtained if (
A() = ((1 ) + + 1) 1/),for all 0 1 and 1. 46
- 47. Arthur CHARPENTIER - cole dt EURIA. 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 Gumbels dependence function A(). 47
- 48. Arthur CHARPENTIER - cole dt EURIA. How to construct much
more copulas ? Using geometric transformationsFrom a given copula
C, cdf of random pair (U, V ), dene the copula of (U, 1 V ), C(U,1V
) (u, v) = u C(u, 1 v) the copula of (1 U, V ), C(1U,V ) (u, v) = v
C(1 u, v) the copula of (1 U, 1 V ), the rotated or survival
copula, C(1U,1V ) (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 - cole dt EURIA. 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
- 50. Arthur CHARPENTIER - cole dt EURIA. 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 18: Using
geometric transformation to generate new copulas. 50
- 51. Arthur CHARPENTIER - cole dt EURIA. 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 19: Using
geometric transformation to generate new copulas. 51
- 52. Arthur CHARPENTIER - cole dt EURIA. 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 20: Using
geometric transformation to generate new copulas. 52
- 53. Arthur CHARPENTIER - cole dt EURIA. Using mixture of
copulasLemma 14. The set of copulas is convex, i.e. if {C , } is a
collection ofcopulas, C(u, v) = C (u, v)d() Ris a copula, where is
a distribution on Thus C = C1 + (1 )C2 denes a copula for all [0,
1].Example 15. Frchet (1951) suggested a mixture of the lower and
the upperbound, C(u, v) = C (u, v) + (1 )C + (u, v), for some [0,
1].Example 16. Mardia (1970) suggested a mixture of the lower, the
upperbound, and the independent copula 2 2 2 + C(u, v) = C (u, v) +
(1 )C (u, v) + C (u, v), [0, 1]. 2 2 53
- 54. Arthur CHARPENTIER - cole dt EURIA. Using distortion
functionsDenition 17. A distortion function is a function h : [0,
1] [0, 1] strictlyincreasing such that h(0) = 0 and h(1) = 1.The
set of distortion function will be denoted H.Note that hH if and
only if h1 H. Given a copula C, dene Ch (u, v) = h1 (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
- 55. Arthur CHARPENTIER - cole dt EURIA.Example 19. if C(u, v) =
uv = C (u, v) (the independent copula), and() = log h(), then Ch
(u, v) = h1 (h(u)h(v)) = 1 ((u) + (v)).Example 20. if h(x) = [1 ex
]/[1 e ] (an exponential distortion), and ifC = C , then 1 (eu
1)(ev 1) Ch (u, v) = log 1 + , e 1which is Frank copula. 55
- 56. Arthur CHARPENTIER - cole dt EURIA. 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 - cole dt EURIA. Monte Carlo and
copulasGeneration of independent variables can be done using a
Random function.Denition 21. Function Random should satisfy the
following properties (i) forall 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=1Thus, one can generate easily random
vectors U = (U1 , ..., Ud ) with independentcomponent. 57
- 58. Arthur CHARPENTIER - cole dt EURIA.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 , . . . , Ud1 ud1 ) P(Ud1 ud1
|U1 u1 , . . . , Ud2 ud2 ) ... P(U3 u3 |U1 u1 , U2 u2 ) P(U2 u2 |U1
u1 ) P(U1 u1 ). 58
- 59. Arthur CHARPENTIER - cole dt EURIA.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]) h0 P(u1 U1 u1 + h, U2 u2 , U3 1, . . . Ud 1) = lim
h0 P(U1 [u1 , u1 + h]) P(U1 u1 + h, U2 u2 , U3 1, . . . Ud 1) P(U1
u1 , U2 u2 , U3 1, . . . = lim h0 P(U1 [u1 , u1 + h]) C(u1 + h, u2
, 1, . . . , 1) C(u1 , u2 , 1, . . . , 1) C = lim = C(u1 , u2 , 1,
. . . , 1). h0 h u1and more generally, k1 P(Uk uk |U1 = u1 , . . .
, Uk1 = uk1 ) = C(u1 , . . . , uk , 1, . . . , 1). u1 . . . uk1
59
- 60. Arthur CHARPENTIER - cole dt EURIA.Thus, U = (U1 , .., Un )
with copula C could be simulated using the followingalgorithm,
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,...,k1 C(|u1 , ...,
uk1 ), uk [1,...,k1 C(|u1 , ..., uk1 )]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 - cole dt EURIA.Example: for Claytons
copula, C(u, v) = (u + v 1)1/ , (U, V ) has jointdistribution C if
and only if U is uniform on on [0, 1] and V |U = u hasconditional
distribution P(V v|U = u) = 2 C(v|u) = (1 + u [v 1])11/ .The
algorithm to generate Claytons 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
)/(1+ 1]u + 11/ . 1 61
- 62. Arthur CHARPENTIER - cole dt EURIA. 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 Claytons 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 Claytons copula. 62
- 63. Arthur CHARPENTIER - cole dt EURIA. 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
- 64. Arthur CHARPENTIER - cole dt EURIA. 400 400 / / . . 200 200
0 0 010 012 014 014 015 610 010 012 014 014 015 610
!is$ri&u$i(n +e - !is$ri&u$i(n +e 7 4 015 2 0 014 !2 010
010 012 014 014 015 610 !2 0 2 4 -ni8(r9 9argins x)dx, one gets rT
C(T, K) = e Q(ST > x)dx, Khence C P Q(ST x) = erT (T, x), or
Q(ST x) = erT (T, x) K Kwhere P denotes the price of a put option.
1 2Consider an option on 2 assets, with payo h(ST , ST ). The price
at time 0 iserT EQ (h(ST , ST )). 1 2 70
- 71. Arthur CHARPENTIER - cole dt EURIA. Copulas in nance: call
on maximum 1 2 1 2Here the payo is h(ST , ST ) = (max{ST , ST } K)+
. The price is then C(T, K) = erT EQ ((max{ST , ST } K)+ ) 1 2 =
erT 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 2hence, if (ST , ST ) has copula C (under
Q), then P 1 P 2 C(T, K) = erT 1 C erT (T, x), erT (T, x) dx. K K K
71
- 72. Arthur CHARPENTIER - cole dt EURIA. Copulas in nance: call
on spreads 1 2 1 2Here the payo is h(ST , ST ) = ([ST ST ] K)+ .
The price is then rT 1 2 rT 2 1C(T, K) = e EQ ((ST ST K)+ ) = e EQ
1(ST + K x ST )dx = erT 2 2 1 Q(K + ST x) Q(ST + K x, ST x} x) dx,
1 2 Q(ST x,ST x+K) 1 2hence, if (ST , ST ) has copula C (under Q),
then rT rT P 2 rT P 1 rT P 2C(T, K) = e e (T, xK)C e (T, x), e (T,
x K) dx. K K K 72
- 73. Arthur CHARPENTIER - cole dt EURIA. Copulas in nance: bonds
on option pricesUsing Tchens inequality, it is possible to derive
bounds for options when thepayo is supermodular. 73
- 74. Arthur CHARPENTIER - cole dt EURIA. Agenda General
introductionModelling correlated risks A short introduction to
copulas Quantifying dependence Statistical inference Agregation
properties 74
- 75. Arthur CHARPENTIER - cole dt EURIA. Natural properties for
dependence measuresDenition 23. is measure of concordance if and
only if satises1. is dened for every pair (X, Y ) of continuous
random variables,2. 1 (X, Y ) +1, (X, X) = +1 and (X, X) = 1,3. (X,
Y ) = (Y, X),4. if X and Y are independent, then (X, Y ) = 0,5. (X,
Y ) = (X, Y ) = (X, Y ),6. if (X1 , Y1 ) P QD (X2 , Y2 ), then (X1
, Y1 ) (X2 , Y2 ),7. if (X1 , Y1 ) , (X2 , Y2 ) , ... is a sequence
of continuous random vectors that converge to a pair (X, Y ) then
(Xn , Yn ) (X, Y ) as n . 75
- 76. Arthur CHARPENTIER - cole dt EURIA.As pointed out in
Scarsini (1984), most of the axioms are self-evident .If is measure
of concordance, then, if f and g are both strictly increasing,
then(f (X), g(Y )) = (X, Y ). Further, (X, Y ) = 1 if Y = f (X)
with f almostsurely strictly increasing, and analogously (X, Y ) =
1 if Y = f (X) with falmost surely strictly decreasing (see
Scarsini (1984)). 76
- 77. Arthur CHARPENTIER - cole dt EURIA. Association measures:
Kendalls and Spearmans Rank correlations can be considered, i.e.
Spearmans dened as 1 1 (X, Y ) = corr(FX (X), FY (Y )) = 12 C(u,
v)dudv 3 0 0and Kendalls dened as 1 1 (X, Y ) = 4 C(u, v)dC(u, v)
1. 0 0 Historical version of those coecientsSpearmans rho was
introduced in Spearman (1904) as (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 - cole dt EURIA.Similarly Kendalls tau
was not dened using copulae, but as the probability ofconcordance,
minus the probability of discordance, i.e. (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 ) (seeNelsen
(1999)). 4QEquivalently, (X, Y ) = 1 2 1) where Q is the number of
inversions n(nbetween the rankings of X and Y (number of
discordance). 78
- 79. Arthur CHARPENTIER - cole dt EURIA. 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
- 80. Arthur CHARPENTIER - cole dt EURIA. The case of the
Gaussian random vectorIf (X, Y ) is a Gaussian random vector with
correlation r, Kruskal then ( (1958)) 6 r 2 (X, Y ) = arcsin and
(X, Y ) = arcsin (r) . 2 80
- 81. Arthur CHARPENTIER - cole dt EURIA. Link between Kendalls
tau and Spearmans rhoNote that Kendalls tau and Spearmans are
linked: it is impossible to have atthe same time 0.4 and = 0.Hence
and satisfy 3 1 1 + 2 2 if 0 2 2 2 + 2 1 1 + 3 if 0. 2 2which yield
the area given below. 81
- 82. Arthur CHARPENTIER - cole dt EURIA. 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 and . 82
- 83. Arthur CHARPENTIER - cole dt EURIA. From Kendalltau to
copula parameters Kendalls 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.0 Gaussian 0.00 0.16 0.31 0.45 0.59 0.71 0.81 0.89 0.95 0.99 1.00
Gumbel 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 + Plackett
1.00 1.57 2.48 4.00 6.60 11.4 21.1 44.1 115 530 + Clayton 0.00 0.22
0.50 0.86 1.33 2.00 3.00 4.67 8.00 18.0 + Frank 0.00 0.91 1.86 2.92
4.16 5.74 7.93 11.4