MRS Copula-GJR-Skewed-t

Transcription

1 JOURNAL OF SYSTEMS ENGINEERING Vol.28 No.1 Feb MRS Copula-GJR-Skewed-t 1,2, 1, 1, 1 (1., ; 2., ) : Copula GJR-Skew-t,4. : ;, ;, Copula,. :;;Copula : F : A : (2013) Hedging of stock index futures based on Markov regime-switching Copula-GJR-Skewed-t model Xie Chi 1,2, Yu Cong 1, Luo Changqing 1, Wang Gangjin 1 (1. College of Business Administration, Hunan University, Changsha , China; 2. Center of Finance and Investment Management, Hunan University, Changsha , China ) Abstract: This paper constructs a GJR-Skew-t model based on Copula functions with Markov regime switching to estimate the minimum variance hedging ratio between the returns of stock index futures and spots in four Asian markets. The empirical results show that the risk mitigation degree of the dynamic hedging models is higher than that of the static models. Based on the analysis of the variance reduction of hedging portfolio, the dynamic models are more effective than other models. Moreover, the proposed Markov regime-switching Copula-GJR-Skewed-t model can gain higher revenues than the traditional models except for the Japanese market, which means that our model contributes to reduce the cost of hedging. Key words: stock index futures; hedging; Markov regime switching Copula function 1,... : ;: : (07AJL005); (2010GXS5B141); (IRT0916); (09YJC630063); (09YBA 037).

2 84 28, , 300,.,. (hedging),,,.,..,garch,.,., GARCH,. Copula,,,., Copula GARCH, GARCH., Copula [1], Copula,, Copula [2].,,Copula. Hsu [3] Copula GARCH,, (CCC)GARCH (DCC)GARCH. Power [4] Copula- GARCH GARCH BEKK GARCH. Lai [5] 5 Copula Copula-TGARCH,5, Copula DCC GARCH. [6], Copula- GARCH. [7] Copula, Copula., Copula. Rodriguez [8], Jondeau [9] Okimoto [10],, Copula (Markov Regime Switching, MRS),MRS Copula 1. Lee [11] Gumbel-Clayton Copula GARCH (RSGC),Copula, Copula., : 1), ; 2),,. Garcia [12] Chollete [13] Bollerslev [14], Engle [15] Pelletier [16], MRS Copula GARCH, MRS Copula. MRS Copula- GARCH, GARCH Copula,.,,, Lee [11]. 1 ()().

3 1 : MRS Copula-GJR-Skewed-t 85 Copula-GARCH,.,, MRS Copula GJR-Skew-t,.,,. 2 MRS Copula 2.1 GARCH, DCC GARCH CCC GARCH.,,,., Copula,,.,Hansen [17] Glosten [18], GJR-skew-t. s t = ϕ s + ϕ 1,s s t 1 + ϕ 2,s f t 1 + ε s,t, (1) f t = ϕ f + ϕ 1,f s t 1 + ϕ 2,f f t 1 + ε f,t, (2) σ 2 s,t = c s + a s,1 ε 2 s,t 1 + b sσ 2 s,t 1 + a s,2k s,t 1 ε 2 s,t 1, (3) σ 2 f,t = c f + a f,1 ε 2 s,t 1 + b f σ 2 f,t 1 + a f,2 k f,t 1 ε 2 s,t 1, (4) ε s,t /σ s,t = z s,t skewed-t(z s η s,ω s ), (5) ε f,t /σ f,t = z f,t skewed-t(z f η f,ω f ), (6) s t t, f t t., i = s f,. ϕ i, ϕ 1,i ϕ 2,i i. σ i,t t i, c i, a i,1, b i a i,2. a i,1 + b i + 0.5a i,2. a i,2 i, a i,2. k i,t 1, ε i,t 1, k i,t 1 = 1;, k i,t 1 = 0. z i,t t, t ( bc ( bz + a η 2 1 ω skewed-t(z η, ω) = ( bc η 2 ) 2 ) η+1/2,z < a b ( ) 2 ) η+1/2 bz + a,z a 1 + ω b, a = 4ω η 2 η 1,b = 1 + 3ω2 a 2 Γ(η + 1/2),c =, η, (4, 30); ω π(η 2)Γ(η/2) ( ), ( 1, 1). 2.2 Copula Copula Copula t Copula, ρ,, Copula. Hsu [3] (7)

4 86 28,, Copula, Lai [5].,,., Copula. 1)Copula Copula, ( c Normal 1 ρ2 ψ 1 (u t ) 2 + ψ 1 (ν t ) 2) 2ρψ 1 (u t ) ψ 1 (ν t ) (u t,v t ;ρ) = exp 1 ρ 2 2(1 ρ 2 ), (8) ψ,ρ ( 1,1). 2) t-copula ρ n, Γ( n+2 c student t (u t,v t ;ρ,n) = (1 ρ 2 ) ) [ Γ( n ) ][ ] n+1 2 Γ( n) Γ( n+1) (1 + a2 t n )(1 + b2 t n ) [ 1 + a2 t + b 2 t 2ρa t b t n(1 ρ 2 ) a t = t 1 n (u t),b t = t 1 n (v t), t 1 n ( ) n t-, ρ ( 1,1). 2.3 ] n+2 2, Patton [19] Copula., ˆξ s ˆξ f ˆξ s arg max T lnf 1 (z s,t Ω t 1 ;ξ s ), t=1 ˆξ f arg max T lnf 2 (z f,t Ω t 1 ;ξ f ). t=1, ˆξ s ˆξ f Copula, (10) Copula 2.4 MRS Copula ˆξ c argmax T t=1 ln c t ( F 1 ( z s,t Ω t 1 ; ˆξ s ),F 2 ( z f,t Ω t 1 ; ˆξ f ) Ω t 1,ξ c ). (10), MRS Copula. MRS Copula,,. MRS Copula, Copula θ t θ t, Copula θ t., MRS Copula Copula., 2.,θ t = {1,2} 2, p ij = Pr(θ t = j θ t 1 = i),i,j = 1,2. p 11 = p,p 22 = q, ( ) p 1 p P =. 1 q q, Copula, Normal Copula c MRS t c MRS t (u t,v t Ω t 1 ;ξ m c,t ) = π 1tc (1) t (u t,v t Ω t 1,ρ 1 ) + (1 π 1t )c (2) t (u t,v t Ω t 1,ρ 2 ), (11) (9)

5 1 : MRS Copula-GJR-Skewed-t 87 u t = F 1 (z s,t Ω t 1 ;ξ s ),v t = F 2 (z f,t Ω t 1 ;ξ f ) z s,t z f,t. π 1t t 1. Gray [20] [ π 1t = Pr(θ t = 1 Ω t 1 ) = p c (1) t 1π 1t 1 c (1) t 1π 1t 1 + c (2) t 1(1 π 1t 1 ) ] [ ] c (2) + (1 q) t 1(1 π 1t 1 ), c (1) t 1π 1t 1 + c (2) t 1(1 π 1t 1 ) c (1) t 1( ) c (2) t 1( )Copula t ρ 1 ρ (11),z s,t z f,t 2.5 f(z s,t,z f,t Ω t 1 ) = c MRS t (u t,v t Ωt 1 ;ξ m c,t )f 1(z s,t Ω t 1 ;ξ s )f 2 (z f,t Ω t 1 ;ξ f ). (13),., h = cov(s,f) var(f). (14) (OLS) 2.,,,,. s t f t t, t, (s t h t 1 f t ) (12) ĥ t = cov t 1(s t,f t ) var t 1 (f t ). (15),. Copula,., σ sf,t = σ s,t σ f,t, (3) (4) (16), z s,t z f,t f(z s,t,z f,t Ω t 1 )drdw. (16) h t = σ sf,t /σ 2 f,t. (17) ,,,.,,.,,. DATASTREAM. 2 OLS Johnson,s t = c + hf t + ε t. h=cov(s, f)/var(f), h OLS. 3 6, OLS CCC GARCH DCC GARCH DCC normal Copula-GJR DCC t Copula-GJRMRS normal Copula-GJR.

6 88 28 (HNGKNGI) 225 (JAPDOWA) (SNGPORI) (TAIWGHT). (HSI) 225 (ONA) (SST) (TAIFEX)., DATASTREAM. 100, r t = 100(ln(P t ) ln(p t 1 )), (18) r t t, P t P t 1 t t , ,,.,. JB, 1%, 3,.,. LM, ARCH, GARCH., GARCH. 1 Table 1 Descriptive statistics of returns J-B P ARCH(5) P N (1) (2), 1, ,, 1. (1) (2) ϕ 1,s, ϕ 2,s, ϕ 1,f, ϕ 2,f 0. Table 2 2 Cross-correlation matrix of first-order lagging : 2/ N, 5%. N 2 490,., 2/ N 5%, 0.04., ADF PP. 3. 3,

7 1 : MRS Copula-GJR-Skewed-t 89 ADF PP 1%,. 3 Table 3 Unit root test ADF PP (0.0001) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) :,. 3.3, Copula,.,, Copula-GARCH.GJR-skew-t, 4. Table 4 4 Estimation of marginal model parameters i = s i = f i = s i = f i = s i = f i = s i = f ϕ i * ** ** ** * ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) c i ** ** ** ** ** ** * * ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a i, ** ** ** ** ** ** ** ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) b i ** ** ** ** ** ** ** ** ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a i, ** ** ** ** ** ** ** ** ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) η i ** ** ** ** ** ** ** ** ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ω i ** * ** ** ** ** ** * ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) : ** 5%, * 10%. 4,, b i + a i, a i,2 1,. a s,2 a f,2 0, t,,., GJR. a s,2 a f,2,. t ω i η i, t., t, : u v., u v DCC Copula MRS Normal Copula, ,.

8 , MRS Normal Copula 1%,, MRS Normal Copula. q 0.9, ( 2). p ,( 1). Table 5 5 DCC Copula Estimation of DCC Copula function parameters DCC t Copula DCC Normal Copula α β n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Table 6 6 MRS Normal Copula Estimation of MRS Normal Copula function parameters p ( ) ( ) ( ) ( ) q ( ) ( ) ( ) ( ) ρ ( ) ( ) ( ) ( ) ρ ( ) ( ) ( ) ( ) (13) (14), MRS Normal Copula-GJR Copula-GJR. 1MRS Normal Copula-GJR Fig. 1 MRS Normal Copula-GJR Optimal hedging ratio of MRS Normal Copula-GJR model 4, 4 MRS Normal Copula-GJR.

9 1 : MRS Copula-GJR-Skewed-t , Ederington, HE 1 = 100(V i p V b p )/V b p. (19), Ederington,,,.,, : HE 2 = R i p R b p, (20) Rp i V p i, i, b, OLS Table 7 7 Comparation of hedging performance OLS MRS Normal Copula DCC t Copula DCC Normal Copula DCC GARCH CCC GARCH OLS MRS Normal Copula DCC t Copula DCC Normal Copula DCC GARCH CCC GARCH OLS MRS Normal Copula DCC t Copula DCC Normal Copula DCC GARCH CCC GARCH OLS MRS Normal Copula DCC t Copula DCC Normal Copula DCC GARCH CCC GARCH ,. 7,.,, DCC GARCH DCC Copula, MRS Normal Copula OLS,. MRS Normal Copula, , OLS ,, MRS Normal Copula., DCC Normal Copula, MRS Normal Copula,OLS. DCC Normal CopulaMRS Normal Copula., OLS, MRS Normal 5, MRS Normal Copula-GJR DCC Normal Copula-GJR DCC t Copula-GJR MRS Normal Copula, DCC Normal Copula DCC t Copula.

10 92 28 Copula. Normal Copula.,, OLS ; MRS Normal Copula OLS, , Copula DCC GARCH, CCC GARCH OLS. MRS Normal Copula DCC t Copula, , OLS, MRS Normal Copula., 225, 225, MRS Copula-GJR-Skewed-t,,,,.,MRS Normal Copula OLS( ), OLS., MRS Normal Copula 3, MRS Normal Copula,:Normal Copula., MRS Normal Copula.,, MRS Normal Copula. 4 MRS Copula GJR-Skew-t, 4, Copula ( DCC, CCC GARCH ), 1) 4,,.,,,. 2), (OLS). MRS Normal Copula,, OLS.,, MRS Normal Copula. 3),, MRS Normal Copula OLS, DCC GARCH CCC GARCH ;, MRS Normal Copula OLS. MRS Normal Copula.,, Copula,.,.,. : [1],,.[J]., 2004, 19(4):

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