Modeling Jump Dependence using L evy copulas Andrea Krajina, - PowerPoint PPT Presentation

Modeling Jump Dependence using L´ evy copulas Andrea Krajina, Institut f¨ ur Matematische Stochastik, G¨ ottingen Joint work with Roger Laeven, Tilburg University and EURANDOM EURANDOM, Eindhoven, August, 2011. 1 / 25

Outline � Introduction – L´ evy process – L´ evy copula � M-Estimator – Definition. Asymptotic properties – Simulation study: Clayton copula � Example: MSCI equity index data (Europe) 2 / 25

L´ evy process � bivariate L´ evy process: a stochastic process Y = ( Y 1 , Y 2 ) such that – Y 0 = (0 , 0) ; – independent and stationary increments; – stochastically continuous. e i � z, Y � � = e φ ( z ) , where � � by L´ evy -Khintchine representation: E φ ( z ) = i � γ, z �− 1 � � e i � z,y � − 1 − i � z, y � 1 {| y | ≤ 1 } � 2 � z, A z � + ν ( dy ) , R 2 \ (0 , 0) evy process: ( γ, A , ν ) , with γ ∈ R 2 , A ∈ R 2 × 2 and a positive any L´ sigma-finite measure ν on R 2 \ { (0 , 0) } . 3 / 25

L´ evy copula � denote R := ( −∞ , ∞ ] 2 is 2 -increasing if � a function F : S → R , S ⊆ R F ( a 1 , a 2 ) + F ( b 1 , b 2 ) − F ( a 1 , b 2 ) − F ( b 1 , a 2 ) ≥ 0 , for all ( a 1 , a 2 ) , ( b 1 , b 2 ) ∈ S with a 1 ≤ b 1 , a 2 ≤ b 2 2 → R is a L´ � a function F : R evy copula if – F ( u 1 , u 2 ) � = ∞ , if ( u 1 , u 2 ) � = ( ∞ , ∞ ) , – F ( u 1 , u 2 ) = 0 , if u i = 0 , for at least one i = 1 , 2 , – F is 2 -increasing – F i ( u ) = u , i = 1 , 2 , u ∈ R 4 / 25

L´ evy copula 2 is the tail integral of a L´ � a function U : R 2 \ { (0 , 0) } → R , S ⊆ R evy process with the L´ evy measure ν if U ( x 1 , x 2 ) = sgn ( x 1 ) sgn ( x 2 ) ν ( I ( x 1 ) × I ( x 2 )) , where � ( x 1 , ∞ ) , if x 1 ≥ 0 , I ( x 1 ) = ( −∞ , x 1 ] , if x 1 < 0 . � L´ evy copula vs. copula 5 / 25

L´ evy copula � Theorem (Kallsen and Tankov) Let F be a bivariate L´ evy copula and U 1 and U 2 the tail integrals of its marginal processes. Then there exists a bivariate L´ evy process Y whose components have tail integrals U 1 and U 1 , and such that U I (( x i ) i ∈ I ) = F I (( U i ( x i )) i ∈ I ) for any non-empty I ⊆ { 1 , 2 } . The L´ evy measure ν is uniquely determined by U 1 , U 2 and F . 6 / 25

L´ evy copula. Limit relation � Theorem (Kallsen and Tankov) Let X = ( X 1 , X 2 ) be a bivariate L´ evy process with L´ evy copula F and tail integrals U 1 and U 2 . Denote by C ( α 1 ,α 2 ) : [0 , 1] 2 → [0 , 1] a copula of t ( − α 1 X 1 , − α 2 X 2 ) , for t ∈ (0 , ∞ ) 2 and α 1 , α 2 ∈ {− 1 , 1 } . Then t → 0 C ( sgn u 1 , sgn u 2 ) F ( u 1 , u 2 ) = lim ( t | u 1 | , t | u 2 | ) sgn u 1 sgn u 2 , t for any ( u 1 , u 2 ) ∈ Ran ( U 1 ) × Ran ( U 2 ) 7 / 25

M-Estimator: Basic Assumptions � bivariate L´ evy process Y observed at n distinct times separated by ∆ n : Y ∆ n , Y 2∆ n , . . . , Y n ∆ n � n increments: Y ∆ n , Y 2∆ n − Y ∆ n , . . . , Y n ∆ n − Y ( n − 1)∆ n � sample X 1 , . . . , X n , where X i = Y i ∆ n − Y ( i − 1)∆ n , with distribution function H and marginals H 1 and H 2 � assume that its L´ evy copula F belongs to a parametric family F ∈ { F θ : θ ∈ Θ ⊆ R p } , 8 / 25

M-Estimator: Nonparametric Estimator � nonparametric estimator of F (Laeven) n �� � R 1 F ( u 1 , u 2 ) := sgn ( u 1 ) sgn ( u 2 )1 i > n + 1 − ku 1 , if u 1 ≥ 0 ˆ � 1 , R 1 k i ≤ k | u 1 | , if u 1 < 0 i =1 � � R 2 i > n + 1 − ku 2 , if u 2 ≥ 0 R 2 i ≤ k | u 2 | , if u 2 < 0 – R j i is the rank of X ij among X 1 j , . . . , X nj , j = 1 , 2 – k ∈ { 1 , 2 , . . . , n } � k = k n → ∞ and k/n → 0 when n → ∞ � ∆ n → 0 and n ∆ n → ∞ when n → ∞ 9 / 25

M-Estimator: Definition � let T > 0 ; denote S T := [ − T, T ] 2 ∩ Ran ( U 1 ) × Ran ( U 2 ) � let g ≡ ( g 1 , . . . , g q ) T : T → R q , q ≥ p , be an integrable function such that ϕ : Θ → R q defined by � ϕ ( θ ) := g ( u ) F ( u ; θ ) d u T is a homeomorphism between Θ and its image ϕ (Θ) � M-estimator ˆ θ n of θ 0 is defined as the minimizer of the criterion function q � 2 �� � � � ˆ Q k,n ( θ ) = g m ( x ) F n ( x ) − F ( x ; θ ) d x T m =1 10 / 25

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 , t − 1 C ( sgn ( u 1 ) , sgn ( u 2 )) ( t | u 1 | , t | u 2 | ) sgn ( u 1 ) sgn ( u 2 ) − F ( u 1 , u 2 ) = O ( t α ) t uniformly on { ( u 1 , u 2 ) ∈ Ran ( U 1 ) × Ran ( U 2 ): u 2 1 + u 2 2 = c } , – for the α from (AN2), k/n − ∆ n = O (( k/n ) 1+ α ) and k = o ( n 2 α/ (1+2 α ) ) ; 11 / 25

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) ˙ ϕ ( θ 0 ) is of full rank; (C5) k = k n 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 Q k,n has a unique P minimizer ˆ θ n , and ˆ → θ 0 , as n → ∞ . θ n 12 / 25

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 , t − 1 C ( sgn ( u 1 ) , sgn ( u 2 )) ( t | u 1 | , t | u 2 | ) sgn ( u 1 ) sgn ( u 2 ) − F ( u 1 , u 2 ) = O ( t α ) t uniformly on { ( u 1 , u 2 ) ∈ Ran ( U 1 ) × Ran ( U 2 ): u 2 1 + u 2 2 = c } , (AN3) for the α from (AN2), k/n − ∆ n = O (( k/n ) 1+ α ) and k = o ( n 2 α/ (1+2 α ) ) ; then as n → ∞ , √ → N ((0 , 0) T , M ( θ 0 )) . d k (ˆ θ n − θ 0 ) (2) 13 / 25

Simulation Study: Clayton L´ evy Copula � Clayton L´ evy copula is given by | u 1 | − ζ + | u 2 | − ζ � − 1 /ζ � F ( u 1 , u 2 ) = ( η 1 { u 1 u 2 ≥ 0 } − (1 − η ) 1 { u 1 u 2 < 0 } ) , with parameters ζ > 0 and η ∈ [0 , 1] � ζ > 0 unknown, η = 1 fixed � L´ evy process: bivariate Poisson jump diffusion d X t = µ d t + σ d W t + J d N t 14 / 25

Simulation Study: Clayton L´ evy Copula � ν ( d x ) = λ 1 f 1 ( x 1 ; θ 1 ) d x 1 + λ 2 f 2 ( x 2 ; θ 2 ) d x 2 + λ 12 f 12 ( x 1 , x 2 ; θ 12 ) d x 1 d x 2 , � model parameters: µ = (0 , 0) , λ + 1 + λ + 12 = λ + 2 + λ + 12 = 6 , θ 1 = θ 2 = 1 / 30 , σ = (0 . 1 , 0 . 1) , ρ = 0 . 4 , three different values for the L´ evy copula parameter ζ ∈ { 0 . 3 , 1 , 3 } � simulations: 50 samples of size n = 7500 , corresponding to T n = 30 and ∆ n = 1 / 250 (e.g. daily returns over 30 years) 15 / 25

Simulation Study: Clayton L´ evy Copula ζ =0.3, with Brownian noise ζ =1, with Brownian noise 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 0.00 0.00 0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 ζ =3, with Brownian noise 0.15 0.10 0.05 0.00 0.00 0.05 0.10 0.15 16 / 25

Simulation Study: Clayton L´ evy Copula Clayton family with ζ =0.3, η =1; M−est. of ζ , rep=50 Clayton family with ζ =0.3, η =1; M−est. of ζ , rep=50 0.42 0.12 T=5 T=10 0.40 0.10 0.38 0.08 RMSE ^ ζ 0.36 0.06 0.34 0.04 0.32 0.02 T=5 0.30 0.00 T=10 20 40 60 80 100 20 40 60 80 100 k k Figure 1: Estimation of ζ = 0 . 3 , for different values of k and T . 17 / 25

Simulation Study: Clayton L´ evy Copula Clayton family with ζ =1, η =1; M−est. of ζ , rep =50 Clayton family with ζ =1, η =1; M−est. of ζ , rep =50 T=5 1.05 T=5 T=10 0.20 T=10 0.15 1.00 RMSE ^ ζ 0.10 0.95 0.05 0.00 0.90 20 40 60 80 100 20 40 60 80 100 k k Figure 2: Estimation of ζ = 1 , for different values of k and T . 18 / 25

Simulation Study: Clayton L´ evy Copula Clayton family with ζ =3, η =1; M−est. of ζ , rep =50 Clayton family with ζ =3, η =1; M−est. of ζ , rep =50 3.0 T=5 T=10 1.0 2.9 0.8 2.8 2.7 0.6 RMSE ^ ζ 2.6 0.4 2.5 0.2 2.4 T=5 T=10 2.3 0.0 20 40 60 80 100 20 40 60 80 100 k k Figure 3: Estimation of ζ = 3 , for different values of k and T . 19 / 25

Simulation Study: Clayton L´ evy 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 Clayton(0.3,1); M− and nonpar. est. of F(1,1)=0.099, rep=50 0.20 T=5 T=10 non.par. 0.18 0.10 0.16 0.08 0.14 0.06 F ( 1 , 1 ) RMSE ^ 0.12 0.04 0.10 0.02 T=5 0.08 T=10 non.par. 0.00 20 40 60 80 100 20 40 60 80 100 k k Figure 4: Estimation of F (1 , 1) = 2 − 1 / 0 . 3 ≈ 0 . 099 , for different values of k and T . 20 / 25

Simulation Study: Clayton L´ evy 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 Clayton(1,1); M− and nonpar. est. of F(1,1)=0.5, rep=50 0.52 T=5 T=10 T=5 non.par. T=10 0.51 non.par. 0.15 0.50 0.49 0.10 F ( 1 , 1 ) RMSE ^ 0.48 0.05 0.47 0.46 0.00 20 40 60 80 100 20 40 60 80 100 k k Figure 5: Estimation of F (1 , 1) = 0 . 5 , for different values of k and T . 21 / 25