Copula và ứng dụng trong tài chính

X, Y, Z, . . .
ξ, η, ζ, . . .
x, y, z, . . .
ξ ξ
F
ξ
(x) = P {ξ < x}.
x F x x (−∞, +∞)
x F (x)
x x
F
ξ
(x) ξ
ξ = (ξ
1
, ξ
2
, . . . , ξ
n
) ξ
i
i = 1, 2, . . . , n
ξ n
ξ
F
ξ
(x

1
, x
2
, . . . , x
n
) = P {ξ
1
< x
1
, ξ
2
< x
2
, . . . , ξ
n
< x
n
}, x
i
∈ R, i = 1, . . . , n .
ξ
i
i = 1, . . . , n
F
ξ
(x
1
, x
2
, . . . , x

n
) = P {ξ
1
< x
1
}.P {ξ
2
< x
2
}. . . P{ξ
n
< x
n
}
= F
ξ
1
(x
1
)F
ξ
2
(x
2
) . . . F
ξ
n
(x
n
).

ξ
i
f(x
1
, x
2
, . . . , x
n
) =
∂
n
F (x
1
, x
2
, . . . x
n
)
∂x
1
∂x
2
. . . ∂x
n
= f
ξ
1
(x
1
).f

ξ
2
(x
2
) . . . f
ξ
n
(x
n
)
f, f
ξ
i
ξ ξ
i
.
(x
1
, x
2
, . . . , x
n
) n
F
n
(x) =
f{x
i
< x}
n

f{x
i
< x} x
i
x x
F
n
(x) x
S
1
, . . . , S
n
R R
[−∞, +∞] H
n DomH = S
1
× ··· × S
n
a ≤ b
a = (a
1
, . . . , a
n
), b = (b
1
, . . . , b
n
) a
k
≤ b

k
k = 1, n B = [a, b] =
[a
1
, b
1
] × ··· × [a
n
, b
n
] n DomH
H B
V
H
(B) =

sgn(c)H( c)
c sgn(c)
c
k
= a
k
k
H B = [a, b]
V
H
(B) = ∆
b
a
H(t) = ∆

b
n
a
n
. . . ∆
b
1
a
1
H(t)
∆
b
k
a
k
H(t) = H(t
1
, . . . , t
k−1
, b
k
, t
k+1
, . . . , t
n
) −H(t
1
, . . . , t
k−1
, a

k
, t
k+1
, . . . , t
n
).
H(x
1
, . . . , x
n
) = P (X
1
≤ x
1
, . . . , X
n
≤ x
n
)
n X
1
, . . . , X
n
V
H
(B) = P (a
1
≤ X
1
≤ b

1
, . . . , a
n
≤ X
n
≤ b
n
).
H n n V
H
(B) ≥ 0
n B DomH
H n DomH =
S
1
× ··· × S
n
S
k
a
k
H
H(t) = 0 t DomH t
k
= a
k
k
S
k
b

k
H
H H
k
DomH
k
= S
k
H
k
(x) = H(b
1
, . . . , b
k−1
, x, b
k+1
, . . . , b
n
) x ∈ S
k
S
1
, . . . , S
k
R H
n S
1
×···×S
n
H

(t
1
, . . . , t
k−1
, x, t
k+1
, . . . , t
n
) ( t
1
, . . . , t
k−1
, y, t
k+1
, . . . , t
n
)
DomH x ≤ y H(t
1
, . . . , t
k−1
, x, t
k+1
, . . . , t
n
) ≤ H(t
1
, . . . , t
k−1
, y, t

k+1
, . . . , t
n
).
S
1
, . . . , S
k
R H
n S
1
×···×S
n
x = (x
1
, . . . , x
n
) y = (y
1
, . . . , y
n
) S
1
× . . . × S
n
|H(x) −H(y)| ≤
n

k=1
|H

k
(x
k
) − H
k
(y
k
)|.
n H
R
n
H H(+∞, . . . , +∞) = 1
n
F
1
, . . . , F
n
N N [0, 1]
N
[0, 1]
I
N
= [0, 1]
N
N
N C
C DomC = I
N
= [0, 1]
N

;
C N
C C
n
C
n
(u) = C(1, . . . , 1, u, 1, . . . , 1) =
u, ∀u ∈ I.
N C N ≥ 3 k C
k N C [0, 1]
N
[0, 1]
u ∈ [0, 1]
N
C(u) = 0 u 0 C(u) = u
k
u u
k
.
a b [0, 1]
N
a
i
≤ b
i
i V
C
([a, b]) ≥ 0.
ImC = I C
F

1
, . . . , F
N
C(F
1
(x
1
), . . . , F
n
(x
n
), . . . , F
N
(x
N
))
F
1
, . . . , F
N
u
n
= F
n
(x
n
)
C N u v [0, 1]
N
|C(v) −C(u)| ≤

N

k=1
|v
k
− u
k
|
C [0, 1]
N
.
F N
F
1
, . . . , F
N
F
F (x
1
, . . . , x
n
, . . . , x
N
) = C(F
1
(x
1
), . . . , F
n
(x

n
), . . . , F
N
(x
N
)).
N
N
F F
F
−1
(t) = inf{x ∈ R|F (x) ≥ t} t ∈ [0, 1] inf ∅ = −∞.
H n
F
1
, . . . , F
n
C C u
[0, 1]
n
C(u
1
, . . . , u
n
) = H(F
−1
1
(u
1
), . . . , F

−1
n
(u
n
)).
F (x
1
, x
2
) = (1+
e
−x
1
+e
−x
2
)
−1
R
2
F
1
(x
1
) =

R
F (x
1
, x

2
)dx
2
= (1+e
−x
1
)
−1
F
2
(x
2
) =

R
F (x
1
, x
2
)dx
1
= (1+e
−x
2
)
−1
C(u
1
, u
2

) = F (F
−1
1
(u
1
), F
−1
2
(u
2
)) =
u
1
u
2
u
1
+ u
2
− u
1
u
2
.
ρ
C
−
C
+

C
⊥
[0, 1]
N
C
−
(u
1
, . . . , u
n
, . . . , u
N
) = max(
N

n=1
u
n
−N + 1, 0)
C
+
(u
1
, . . . , u
n
, . . . , u
N
) = min(u
1
, . . . , u

n
, . . . , u
N
)
C
⊥
(u
1
, . . . , u
n
, . . . , u
N
) =
N

n=1
u
n
C
−
, C
+
C
⊥
C
+
, C
⊥
N N ≥ 2
C

−
N ≥ 3
n [1/2, 1]
n
⊂ [0, 1]
n
V
C
−
([1/2, 1]
n
) = max(1 + ··· + 1 − n + 1, 0)
− n max(1/2 + 1 + ··· + 1 − n + 1, 0)
+

n
2

max(1/2 + 1/2 + 1 + ··· + 1 − n + 1, 0)
···
+ max(1/2 + ··· + 1/2 − n + 1, 0)
= 1 − n/2 + 0 + ···+ 0.
C
−
n ≥ 3
C
1
C
2
C

2
C
1
C
1
≺ C
2
C
2
 C
1
∀(u
1
, . . . , u
n
, . . . , u
N
) ∈ I
N
, C
1
(u
1
, . . . , u
n
, . . . , u
N
) ≤ C
2
(u

1
, . . . , C
n
, . . . , C
N
)
∀(u
1
, . . . , u
n
, . . . , u
N
) ∈ I
N
, C
1
(u
1
, . . . , u
n
, . . . , u
N
) ≤ C
2
(u
1
, . . . , u
n
, . . . , u
N

)
C
1
(u
1
, u
2
) ≤ C
2
(u
1
, u
2
) ⇔ 1 − u
1
−u
2
+ C
1
(u
1
, u
2
) ≤ 1 − u
1
−u
2
+ C
2
(u

1
, u
2
)
⇔ C
1
(u
1
, u
2
) ≤ C(u
1
, u
2
)
C n
C (U
1
, . . . , U
n
)
T
C C(u
1
, . . . , u
n
) = P{U
1
> u
1

, . . . , U
n
> u
n
}
n C u ∈ [0, 1]
n
C
−
(u) ≤ C(u) ≤ C
+
(u)
n ≥ 3 u ∈ [0, 1]
n
n C
u
C(u) = C
−
(u)
C(u
1
, u
2
, ρ) = Φ
ρ
(Φ
−1
(u
1
); Φ

−1
(u
2
))
C
−
= C
ρ=−1
≺ C
ρ<0
≺ C
ρ=0
= C
⊥
≺ C
ρ>0
≺ C
ρ=1
= C
+
{(u
1
, u
2
) ∈ I
2
|C(u
1
, u
2

) = C}
C(u
1
, u
2
) =
1
α
ln

1 +
(exp(αu
1
) − 1)(exp(αu
2
) − 1)
(exp(α) − 1)

α ∈ R

α
α −∞ +∞
c
c(u
1
, . . . , u
n
, . . . , u
N

) =
∂C(u
1
, . . . , u
n
, . . . , u
N
)
∂u
1
. . . ∂u
n
. . . ∂u
N
.
f N F
f(x
1
, . . . , x
n
, . . . , x
N
) = c(F
1
(x
1
), . . . , F
n
(x
n

), . . . , F
n
(x
N
))
N

n=1
f
n
(x
n
)
f
n
F
n
X
1
, . . . , X
n
F
1
, . . . , F
n
H (X
1
, . . . , X
n
)

T
C C
(X
1
, . . . , X
n
)
T
H(x
1
, . . . , x
n
) = P{X
1
≤ x
1
, . . . , X
n
≤ x
n
} = C(F
1
(x
1
), . . . , F
n
(x
n
))
X

i
→ F (X
i
)
X
1
, . . . , X
n
H(x
1
, . . . , x
n
) = F
1
(x
1
) . . . F
n
(x
n
)
x
1
, . . . , x
n
R.
(X
1
, . . . , X
n

)
T
C X
1
, . . . , X
n
C = C
⊥
.
X
α RanX
Range α(X)
(X
1
, . . . , X
n
)
T
C α
1
, . . . , α
n
RanX
1
, . . . , RanX
n
(α
1
(X
1

), . . . , α
n
(X
n
))
T
C
F
1
, . . . , F
n
X
1
, . . . , X
n
G
1
, . . . , G
n
α
1
(X
1
), . . . , α
n
(X
n
).
X
1

, . . . , X
n
C (α
1
(X
1
), . . . , α
n
(X
n
))
T
C
α
α
k
k G
k
(x) = P{α
k
(X
k
) ≤ x} = P{X
k
≤ α
−1
k
(x)} = F
k
(α

−1
k
(x))
x R
C
α
(G
1
(x
1
), . . . , G
n
(x
n
)) = P{α
1
(X
1
) ≤ x
1
, . . . , α
n
(X
n
) ≤ x
n
}
= P{X
1
≤ α

−1
1
(x
1
), . . . , X
n
≤ α
−1
n
(x
n
)}
= C(F
1
(α
−1
1
(x
1
)), . . . , F
n
(α
−1
n
(x
n
)))
= C(G
1
(x

1
), . . . , G
n
(x
n
))
X
1
, . . . , X
n
RanG
1
= . . . = RanG
n
= [0, 1] C
α
= C
[0, 1]
n
.
C
n
ˆ
C n
(X
1
, . . . , X
n
)
T

C
X
1
, ,X
n
α
1
, . . . , α
n
RanX
1
, . . . , RanX
n
(α
1
(X
1
), . . . , α
n
(X
n
))
T
C
α
1
(X
1
), ,α
n

(X
n
)
α
k
k k = 1
C
α
1
(X
1
), ,α
n
(X
n
)
(u
1
, u
2
, . . . , u
n
) = C
α
2
(X
2
), ,α
n
(X

n
)
(u
2
, . . . , u
n
)
− C
X
1
,α
2
(X
2
), ,α
n
(X
n
)
(1 − u
1
, u
2
, . . . , u
n
).
X
1
, . . . , X
n

F
1
, . . . , F
n
α
1
(X
1
), . . . , α
n
(X
n
) G
1
, . . . , G
n
C
α
1
(X
1
),α
2
(X
2
), ,α
n
(X
n
)

(G
1
(X
1
), . . . , G
n
(X
n
))
= P{α
1
(X
1
) ≤ x
1
, . . . , α
n
(X
n
) ≤ x
n
}
= P{X
1
> α
−1
1
(x
1
), α

2
(X
2
) ≤ x
2
, . . . , α
n
(X
n
) ≤ x
n
}
= P{α
2
(X
2
) ≤ x
2
, . . . , α
n
(X
n
) ≤ x
n
}
− P{X
1
≤ α
−1
1

(x
1
), α
2
(X
2
) ≤ x
2
, . . . , α
n
(X
n
) ≤ x
n
}
= C
α
2
(X
2
), ,α
n
(X
n
)
(G
2
(X
2
), . . . , G

n
(X
n
))
− C
X
1
,α
2
(X
2
), ,α
n
(X
n
)
(F
1
(α
−1
1
(x
1
)), G
2
(x
2
), . . . , G
n
(x

n
))
= C
α
2
(X
2
), ,α
n
(X
n
)
(G
2
(X
2
), . . . , G
n
(X
n
))
− C
X
1
,α
2
(X
2
), ,α
n

(X
n
)
(1 − G
1
(x
1
), G
2
(x
2
), . . . , G
n
(x
n
)).
α
1
RanX
1
α
2
RanX
2
C
α
1
(X
1
),α

2
(X
2
)
(u
1
, u
2
) = u
2
− C
X
1
,α
2
(X
2
)
(1 − u
1
, u
2
)
= u
2
− C
X
1
,X
2

(1 − u
1
, u
2
).
α
1
α
2
C
α
1
(X
1
),α
2
(X
2
)
(u
1
, u
2
) = u
2
− C
X
1
,α
2

(X
2
)
(1 − u
1
, u
2
)
= u
2
− (1 − u
1
− C
X
1
,X
2
(1 − u
1
, 1 − u
2
))
= u
1
+ u
2
− 1 + C
X
1
,X

2
(1 − u
1
, 1 − u
2
).
(X
1
, X
2
)
T
ˆ
C
C
α
1
(X
1
),α
2
(X
2
)
H(x
1
, x
2
) = P{X
1

> x
1
, X
2
> x
2
} =
ˆ
C(F
1
(x
1
), F
2
(x
2
)).
n U(0, 1)
C C(u
1
, . . . , u
n
) =
ˆ
C(1 −u
1
, . . . , 1 −u
n
).
(X, Y )

T
(X, Y )
T
ρ(X, Y ) =
Cov(X, Y )

V ar(X)

V ar(Y )
Cov(X, Y ) = E(XY )−E(X)E(Y ) (X, Y )
T
V ar(X) V ar( Y ) X Y.
Y = aX + b a ∈ R\{ 0} b ∈ R
|ρ(X, Y )| = 1 −1 < ρ(X, Y ) < 1
ρ(αX + β, γY + δ) = sign(αγ)ρ(X, Y )
α, γ ∈ R\{ 0}
δ ∈ R
A, B m × n a, b ∈ R
m
X, Y n
Cov(AX + a, BY + b) = ACov(X, Y )B
T
α ∈ R
V ar(α
T
X) = α
T
Cov(X)α
Cov(X) := Cov(X, X)
(x, y)

T
(˜x, ˜y)
T
(X, Y )
T
(x, y)
T
(˜x, ˜y)
T
(x − ˜x)(y − ˜y) > 0
(x − ˜x)(y − ˜y) < 0
(X, Y )
T
(
˜
X,
˜
Y )
T
H
˜
H F
X
˜
X
G Y
˜
Y
C
˜

C
(X, Y )
T
(
˜
X,
˜
Y )
T
H(x, y) = C(F (x), G(y))
˜
H(x, y) =
˜
C(F (x), G(y))
Q
(X, Y )
T
(
˜
X,
˜
Y )
T
Q = P{(X −
˜
X)(Y −
˜
Y ) > 0}− P{(X −
˜
X)(Y −

˜
Y ) < 0}
Q = Q(C,
˜
C) = 4

[0,1]
2
˜
C(u, v)dC(u, v) −1.
P{(X −
˜
X)(Y −
˜
Y ) < 0} =
1 −P{(X −
˜
X)(Y −
˜
Y ) > 0}
Q = 2P{(X −
˜
X)(Y −
˜
Y ) > 0} − 1
P{(X −
˜
X)(Y −
˜
Y ) > 0} = P{X >

˜
X, Y >
˜
Y }+ P{X <
˜
X, Y <
˜
Y }
P{X >
˜
X, Y >
˜
Y } = P{
˜
X < X,
˜
Y < Y }
=

R
2
P{
˜
X < x,
˜
Y < y}dC(F (x), G(y))
=

R
2

˜
C(F (x), G(y)) d(F (x), G(y)).
u = F (x) v = G(y)
P{X >
˜
X, Y >
˜
Y } =

[0,1]
2
˜
C(u, v) dC(u, v).
P{X <
˜
X, Y <
˜
Y } =

R
2
{
˜
X > x,
˜
Y > y} dC(F (x), G(y))
=

R
2

{1 − F (x) − G(y) +
˜
C(F (x), G(y))} dC(F (x), G(y))
=

[0,1]
2
{1 − u − v +
˜
C(u, v)} dC(u, v).
C (U, V )
T
U(0, 1) E(U) = E(V ) = 1/2
P{X <
˜
X, Y <
˜
Y } = 1 −
1
2
−
1
2
+

[0,1]
2
˜
C(u, v) dC(u, v) =


[0,1]
2
˜
C(u, v) dC(u, v).
P{(X −
˜
X)(Y −
˜
Y ) > 0} = 2

[0,1]
2
˜
C(u, v) dC(u, v)
C,
˜
C, Q
Q Q(C,
˜
C) = Q(
˜
C, C);
Q C ≺ C

Q(C,
˜
C) ≤ Q(C

,
˜

C);
Q Q(C,
˜
C) =
Q(
ˆ
C,
ˆ
˜
C)
κ
X
1
X
2
C
κ X
1
, X
2
−1 = κ
X,−X
≤ κ
C
≤ κ
X,X
= 1;
κ
X
1

,X
2
= κ
X
2
,X
1
;
X
1
, X
2
κ
X
1
,X
2
= κ
C
⊥
= 0;
κ
−X
1
,X
2
= κ
X
2
,−X

1
= −κ
X
1
,X
2
;
C
1
≺ C
2
κ
C
1
≤ κ
C
2
;
{ (X
1,n
; X
2,n
)} C
n
{C
n
} C lim
n→∞
κ
C

n
= κ
C
.
κ
(X
1
, , X
n
, , X
N
)
C
X
1
, ,X
n
, ,X
N
= C
h
1
(X
1
), ,h
n
(X
n
), ,h
N

(X
N
)
∂
x
h
n
(x) > 0.
τ ρ γ
τ = 4

I
2
C(u
1
, u
2
)dC(u
1
, u
2
) − 1
ρ = 12

I
2
u
1
u
2

dC(u
1
, u
2
) − 3
γ = 2

I
2
(|u
1
+ u
2
−1| − |u
1
− u
2
|)dC(u
1
, u
2
)
τ ρ
τ, ρ γ
τ = 1 − 4

I
2
∂
u

1
C(u
1
, u
2
)∂
u
2
C(u
1
, u
2
)du
1
du
2
ρ = 12

I
2
C(u
1
, u
2
)du
1
du
2
− 3
γ = 4


I
(C( u, u) + C(u, 1 − u) − u)du
τ ρ
τ, ρ γ
[−1, 1]
[0, 1].
τ, ρ γ
δ
X
1
X
2
δ X
1
X
2
0 = δ
C
⊥ ≤ δ
C
≤ δ
C
+
= 1;
δ
X
1
,X
2

= δ
X
2
,X
1
;
δ
X
1
,X
2
= δ
C
⊥ = 0 X
1
X
2
δ
X
1
,X
2
= δ
C
+
= 1
X
1
X
2

h
1
h
2
ImX
1
ImX
2
δ
h
1
(X
1
).h
2
(X
2
)
= δ
X
1
,X
2
;
{(X
1,n
, X
2,n
)} C
n

C lim
n→∞
δ
C
n
= δ
C
.
σ = 12

I
2
|C(u
1
, u
2
) − C
⊥
(u
1
, u
2
)|du
1
du
2
Φ
2
= 90


I
2
|C(u
1
, u
2
) − C
⊥
(u
1
, u
2
)|
2
du
1
du
2
.
σ Φ
2
X
1
X
2
P{X
1
> x
1
, X

2
> x
2
} ≥ P{X
1
> x
1
}P{X
2
> x
2
}.
X
1
X
2
C  C
⊥
.
C
lim
u→1
C(u, u)
1 − u
= λ
C λ ∈ (0, 1]
λ = 0 λ
ˆ
C C
ˆ

C(u
1
, u
2
) = u
1
+ u
2
− 1 + C(1 − u
1
, 1 − u
2
)
C U(0, 1)
C
C(u
1
, u
2
) = 1 − u
1
− u
2
+ C(u
1
, u
2
)
C(u
1

, u
2
) =
ˆ
C(1 − u
1
, 1 − u
2
).
lim
u→1
C(u, u)
1 − u
= lim
u→1
ˆ
C(1 −u, 1 − u)
1 − u
= lim
u→0
ˆ
C(u, u)
u
.
C
ˆ
C.
C
ˆ
C.

λ
λ(u) = P{U
1
> u|U
2
> u} =
C(u,u)
1−u
. λ(u)
λ(u)
λ(u)
ρ = 1 λ = 0 ρ < 1
λ(u)
ρ = 1
λ(u) ν = 1
λ(u) ν = 5
C
θ
(u, v) = exp(−[(−ln u)
θ
+ (−ln v)
θ
)]
1
θ
).
θ ≥ 1 C
θ
λ > 1
λ = 1

C(u, u)
1 − u
=
1 − 2u + C(u, u)
1 − u
=
1 − 2u + exp(2
1/θ
ln u)
1 − u
=
1 − 2u + u
2
1/θ
1 − u
,
lim
u→1
C(u, u)
1 − u
= 2 − lim
u→1
2
1/θ
u
2
1/θ
−1
= 2 − 2
1/θ

.
X n µ ∈ R
n
Σ n ×n ϕ
X−µ
(t) X −µ
t
T
Σt ϕ
X−µ
(t) = φ(t
T
Σt) X
µ, Σ φ X ∼ E
n
(µ, Σ, φ).
n = 1
φ
X ∼ E
n
(µ, Σ, φ) rank(Σ) = k
R ≥ 0 U k
{z ∈ R
k
|z
T
z = 1} A n × k AA
T
= Σ
X =

d
µ + RAU.
n X ∼ N
n
(0, I
n
) X
i
∼
N(0, 1), i = 1, . . . , n X
i
exp(−t
2
i
/2)
X
exp{−
1
2
(t
2
1
+ ··· + t
2
n
)} = exp{−
1
2
t
T

t}.
X ∼ E
n
(0, I
n
, φ) φ(u) = exp(−u/2).
X ∼ E
n
(µ, Σ, φ) Σ X
0 < V ar(X
i
) < ∞ X
X ∼ N
n
(µ, Σ)
X ∼ E
n
(µ, Σ, φ)
X |Σ|
−1/2
g((X − µ
T
)Σ
−1
(X − µ)) g
R
n
. X E
n
(µ, Σ, φ)

µ Σ φ
X ∼ E
n
(µ, Σ, φ) X ∼ E
n
(µ
∗
, Σ
∗
, φ
∗
)
µ
∗
= µ, Σ
∗
= cΣ, φ
∗
(.) = φ(./c)
c > 0.
Cov(X) = Σ
Cov(X) = Cov(µ + RAU) = AE(R
2
)Cov(U)A
T
E(R
2
) < ∞ Y ∼ N
n
(0, I

n
). Y =
d
||Y ||U ||Y ||
U ||Y ||
2
∼ χ
2
n
E(||Y ||
2
) = n. Cov(Y ) = I
n
U R
n
Cov(U) = I
n
/n
Cov(X) = AA
T
E(R
2
)/n φ
∗
(s) = φ(s/c)
c = E(R
2
)/n Cov(X) = Σ
µ, Σ φ φ Cov(X) = Σ Cov(X)
Cov(X) X

E(X), Cov(X)
X ∼ E
n
(µ, Σ, φ) B q ×n b ∈ R

b + BX ∼ E
q
(b + Bµ, BΣB
T
, φ).
b + BX
b + BX =
d
b + Bµ + RBAU.
X, µ Σ
X =

X
1
X
2

, µ =

µ
1
µ
2

, Σ =


Σ
11
Σ
12
Σ
21
Σ
22

X
1
µ
1
r × 1 Σ
11
r × r
X ∼ E
n
(µ, Σ, φ)
X
1
∼ E
r
(µ
1
, Σ
11
, φ), X
2

∼ E
n−r
(µ
2
, Σ
22
, φ).
f(x
T
x)