Modeling Jump Dependence using Lévy copulas
Andrea Krajina, Institut für Matematische Stochastik, Göttingen
Joint work with
Roger Laeven, Tilburg University and EURANDOM
EURANDOM, Eindhoven, August, 2011.
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Outline
Introduction
– Lévy process
– Lévy copula
M-Estimator
– Definition. Asymptotic properties
– Simulation study: Clayton copula
Example: MSCI equity index data (Europe)
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Lévy process
bivariate Lévy process: a stochastic process Y = (Y1 , Y2 ) such that
– Y0 = (0, 0);
– independent and stationary increments;
– stochastically continuous.
= eφ(z) , where
by Lévy -Khintchine representation: E
Z
1
ihz,yi
φ(z) = i hγ, zi− hz, Azi+
e
− 1 − i hz, yi 1{|y| ≤ 1} ν(dy),
2
R2 \(0,0)
eihz,Yi
any Lévy process: (γ, A, ν), with γ ∈ R2 , A ∈ R2×2 and a positive
sigma-finite measure ν on R2 \ {(0, 0)}.
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Lévy copula
denote R := (−∞, ∞]
a function F : S → R, S ⊆ R is 2-increasing if
2
F (a1 , a2 ) + F (b1 , b2 ) − F (a1 , b2 ) − F (b1 , a2 ) ≥ 0,
for all (a1 , a2 ), (b1 , b2 ) ∈ S with a1 ≤ b1 , a2 ≤ b2
2
a function F : R → R is a Lévy copula if
– F (u1 , u2 ) 6= ∞, if (u1 , u2 ) 6= (∞, ∞),
– F (u1 , u2 ) = 0, if ui = 0, for at least one i = 1, 2,
– F is 2-increasing
– F i (u) = u, i = 1, 2, u ∈ R
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Lévy copula
2
a function U : R2 \ {(0, 0)} → R, S ⊆ R is the tail integral of a Lévy
process with the Lévy measure ν if
U (x1 , x2 ) = sgn(x1 )sgn(x2 )ν (I(x1 ) × I(x2 )) , where
(
(x1 , ∞), if x1 ≥ 0,
I(x1 ) =
(−∞, x1 ], if x1 < 0.
Lévy copula vs. copula
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Lévy copula
Theorem (Kallsen and Tankov)
Let F be a bivariate Lévy copula and U1 and U2 the tail integrals of its
marginal processes. Then there exists a bivariate Lévy process Y whose
components have tail integrals U1 and U1 , and such that
U I ((xi )i∈I ) = F I ((Ui (xi ))i∈I )
for any non-empty I ⊆ {1, 2}. The Lévy measure ν is uniquely
determined by U1 , U2 and F .
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Lévy copula. Limit relation
Theorem (Kallsen and Tankov)
Let X = (X1 , X2 ) be a bivariate Lévy process with Lévy copula F and
(α ,α )
tail integrals U1 and U2 . Denote by Ct 1 2 : [0, 1]2 → [0, 1] a copula of
(−α1 X1 , −α2 X2 ), for t ∈ (0, ∞)2 and α1 , α2 ∈ {−1, 1}. Then
(sgnu1 ,sgnu2 )
F (u1 , u2 ) = lim Ct
t→0
(t|u1 |, t|u2 |)sgnu1 sgnu2 ,
for any (u1 , u2 ) ∈ Ran(U1 ) × Ran(U2 )
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M-Estimator: Basic Assumptions
bivariate Lévy process Y observed at n distinct times separated by ∆n :
Y∆n , Y2∆n , . . . , Yn∆n
n increments: Y∆n , Y2∆n − Y∆n , . . . , Yn∆n − Y(n−1)∆n
sample X1 , . . . , Xn , where Xi = Yi∆n − Y(i−1)∆n , with distribution
function H and marginals H1 and H2
assume that its Lévy copula F belongs to a parametric family
F ∈ {Fθ : θ ∈ Θ ⊆ Rp },
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M-Estimator: Nonparametric Estimator
nonparametric estimator of F (Laeven)
1
F̂ (u1 , u2 ) := sgn(u1 )sgn(u2 )
k
n
X
i=1
1
((
Ri1
Ri1
)
> n + 1 − ku1 , if u1 ≥ 0
,
≤ k|u1 |,
if u1 < 0
)
(
2
Ri > n + 1 − ku2 , if u2 ≥ 0
if u2 < 0
Ri2 ≤ k|u2 |,
j
– Ri is the rank of Xij among X1j , . . . , Xnj , j = 1, 2
– k ∈ {1, 2, . . . , n}
k = kn → ∞ and k/n → 0 when n → ∞
∆n → 0 and n∆n → ∞ when n → ∞
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M-Estimator: Definition
let T > 0; denote ST := [−T, T ]2 ∩ Ran(U1 ) × Ran(U2 )
let g ≡ (g1 , . . . , gq )T : T → Rq , q ≥ p, be an integrable function such
that ϕ : Θ → Rq defined by
Z
g(u)F (u; θ) du
ϕ(θ) :=
T
is a homeomorphism between Θ and its image ϕ(Θ)
M-estimator θ̂n of θ0 is defined as the minimizer of the criterion
function
Qk,n (θ) =
q Z
X
m=1
T
2
gm (x) F̂n (x) − F (x; θ) dx
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M-Estimator: Asymptotic Properties
existence, uniqueness and consistency
asymptotic normality
conditions:
– there exists α ≥ 1 such that k/n − ∆n = o (k/n)α ;
– F has continuous first derivatives F 1 and F 2 ,
– there exist α > 0 and c > 0 such that, as t → 0,
(sgn(u1 ),sgn(u2 ))
t−1 Ct
(t|u1 |, t|u2 |)sgn(u1 )sgn(u2 ) − F (u1 , u2 ) = O(tα )
uniformly on {(u1 , u2 ) ∈ Ran(U1 ) × Ran(U2 ) : u21 + u22 = c},
– for the α from (AN2), k/n − ∆n = O((k/n)1+α ) and
k = o(n2α/(1+2α) );
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M-Estimator: Consistency
Theorem 1 (Consistency) If the following conditions are satisfied,
(C1) Θ is open,
(C2) ϕ is a homeomorphism from Θ to ϕ(Θ),
(C3) ϕ is a twice continuously differentiable,
(C4) ϕ̇(θ0 ) is of full rank;
(C5) k = kn is an intermediate sequence: k → ∞ and k/n → 0, as n → ∞,
(C6) ∆n is such that ∆n → 0 and n∆n → ∞,
(C7) there exists α ≥ 1 such that k/n − ∆n = o (k/n)α ;
then with probability tending to one the criterion function Qk,n has a unique
P
minimizer θ̂n , and θ̂n → θ0 , as n → ∞.
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M-Estimator: Asymptotic Normality
Theorem 2 (Asymptotic normality) If in addition to the assumptions
(C1)-(C6) of Theorem 1 the following assumptions hold,
(AN1) F has continuous first derivatives F (1) and F (2) ,
(AN2) there exist α > 0 and c > 0 such that, as t → 0,
(sgn(u1 ),sgn(u2 ))
t−1 Ct
(t|u1 |, t|u2 |)sgn(u1 )sgn(u2 ) − F (u1 , u2 ) = O(tα )
uniformly on {(u1 , u2 ) ∈ Ran(U1 ) × Ran(U2 ) : u21 + u22 = c},
(AN3) for the α from (AN2), k/n − ∆n = O((k/n)1+α ) and k = o(n2α/(1+2α) );
then as n → ∞,
√
d
k(θ̂n − θ0 ) → N ((0, 0)T , M (θ0 )).
(2)
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Simulation Study: Clayton Lévy Copula
Clayton Lévy copula is given by
−1/ζ
F (u1 , u2 ) = |u1 |−ζ + |u2 |−ζ
(η1{u1 u2 ≥ 0} − (1 − η)1{u1 u2 < 0}) ,
with parameters ζ > 0 and η ∈ [0, 1]
ζ > 0 unknown, η = 1 fixed
Lévy process: bivariate Poisson jump diffusion
dXt = µdt + σdWt + JdNt
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Simulation Study: Clayton Lévy Copula
ν(dx) = λ1 f1 (x1 ; θ1 )dx1 + λ2 f2 (x2 ; θ2 )dx2 + λ12 f12 (x1 , x2 ; θ12 )dx1 dx2 ,
model parameters:
+
+
+
µ = (0, 0), λ+
+
λ
=
λ
+
λ
1
12
2
12 = 6, θ1 = θ2 = 1/30,
σ = (0.1, 0.1), ρ = 0.4,
three different values for the Lévy copula parameter ζ ∈ {0.3, 1, 3}
simulations: 50 samples of size n = 7500, corresponding to Tn = 30 and
∆n = 1/250 (e.g. daily returns over 30 years)
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Simulation Study: Clayton Lévy Copula
ζ=1, with Brownian noise
0.00
0.00
0.05
0.05
0.10
0.10
0.15
0.15
0.20
0.20
0.25
ζ=0.3, with Brownian noise
0.10
0.15
0.20
0.25
0.00
0.05
0.10
0.15
0.10
0.15
ζ=3, with Brownian noise
0.05
0.05
0.00
0.00
0.00
0.05
0.10
0.15
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Simulation Study: Clayton Lévy Copula
0.12
Clayton family with ζ=0.3, η=1; M−est. of ζ, rep=50
T=5
T=10
0.08
RMSE
0.04
0.06
T=5
T=10
0.00
0.02
0.36
0.34
0.32
0.30
^
ζ
0.38
0.10
0.40
0.42
Clayton family with ζ=0.3, η=1; M−est. of ζ, rep=50
20
40
60
k
80
100
20
40
60
80
100
k
Figure 1: Estimation of ζ = 0.3, for different values of k and T .
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Simulation Study: Clayton Lévy Copula
Clayton family with ζ=1, η=1; M−est. of ζ, rep =50
T=5
T=10
0.20
T=5
T=10
RMSE
^
ζ
0.90
0.00
0.05
0.95
0.10
1.00
0.15
1.05
Clayton family with ζ=1, η=1; M−est. of ζ, rep =50
20
40
60
k
80
100
20
40
60
80
100
k
Figure 2: Estimation of ζ = 1, for different values of k and T .
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Simulation Study: Clayton Lévy Copula
Clayton family with ζ=3, η=1; M−est. of ζ, rep =50
3.0
Clayton family with ζ=3, η=1; M−est. of ζ, rep =50
T=5
T=10
0.6
0.0
2.3
2.4
0.2
2.5
0.4
2.6
^
ζ
RMSE
2.7
2.8
0.8
2.9
1.0
T=5
T=10
20
40
60
k
80
100
20
40
60
80
100
k
Figure 3: Estimation of ζ = 3, for different values of k and T .
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Simulation Study: Clayton Lévy Copula
jump dependence coefficient: F (1, 1) = 2−1/0.3 ≈ 0.099
Clayton(0.3,1); M− and nonpar. est. of F(1,1)=0.099, rep=50
0.10
T=5
T=10
non.par.
0.08
0.00
0.08
0.02
0.10
0.04
0.06
RMSE
0.14
0.12
^
F(1, 1)
0.16
0.18
0.20
Clayton(0.3,1); M− and nonpar. est. of F(1,1)=0.099, rep=50
20
40
60
k
80
100
T=5
T=10
non.par.
20
40
60
80
100
k
Figure 4: Estimation of F (1, 1) = 2−1/0.3 ≈ 0.099, for different values of k
and T .
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Simulation Study: Clayton Lévy Copula
jump dependence coefficient: F (1, 1) = 2−1/1 = 0.5
Clayton(1,1); M− and nonpar. est. of F(1,1)=0.5, rep=50
T=5
T=10
non.par.
T=5
T=10
non.par.
0.10
RMSE
0.49
0.00
0.46
0.47
0.05
0.48
^
F(1, 1)
0.50
0.15
0.51
0.52
Clayton(1,1); M− and nonpar. est. of F(1,1)=0.5, rep=50
20
40
60
k
80
100
20
40
60
80
100
k
Figure 5: Estimation of F (1, 1) = 0.5, for different values of k and T .
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Simulation Study: Clayton Lévy Copula
jump dependence coefficient: F (1, 1) = 2−1/3 ≈ 0.794
Clayton(3,1); M− and nonpar. est. of F(1,1)=0.794, rep=50
Clayton(3,1); M− and nonpar. est. of F(1,1)=0.794, rep=50
0.10
RMSE
0.77
T=5
T=10
non.par.
0.00
0.74
0.75
0.05
0.76
^
F(1, 1)
0.78
0.79
0.15
T=5
T=10
non.par.
20
40
60
k
80
100
20
40
60
80
100
k
Figure 6: Estimation of F (1, 1) = 2−1/3 ≈ 0.794, for different values of k and
T.
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Example: MSCI equity data; Europe
we consider log-returns from the price index for the following pairs of
countries:
– UK-Germany, time period Jan 2, 1980 - Jun 30, 2011, n = 8127
– UK-Portugal, time period Jan 4, 1988 - June 30, 2011, n = 6129
– UK-Greece, time period Jan 4, 1988 - June 30, 2011, n = 6129
– summary statistics: mean≈ 0, std.dev 0.012 − 0.02
UK-Germany
UK-Portugal
UK-Greece
ζ̂, k = 30
0.885
0.545
0.439
F (1, 1) = 2−1/ζ̂
0.457
0.280
0.206
F̂ (1, 1)
0.5
0.33
0.23
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Example: MSCI equity data; Europe
UK − Portugal
0.05
−0.05
0.00
Portugal
0.00
−0.05
−0.10
−0.10
−0.05
0.00
0.05
0.10
−0.10
−0.05
UK
0.00
0.05
0.10
UK
Greece
0.05
0.10
0.15
UK − Greece
0.00
−0.10
−0.05
−0.15
−0.10
Germany
0.05
0.10
0.10
UK − Germany
−0.10
−0.05
0.00
0.05
0.10
UK
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Example: MSCI equity data; Europe
UK − Portugal
0.0
0.0
0.2
0.2
0.4
0.4
^
ζ
^
ζ
0.6
0.6
0.8
0.8
UK − Germany
40
60
80
100
20
40
k
60
80
100
k
0.0
0.1
0.2
0.3
0.4
0.5
UK − Greece
^
ζ
20
20
40
60
80
100
k
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