Part 1 GARCH () ( ) /24, p.2/93
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1 基盤研究 A 統計科学における数理的手法の理論と応用 ( 研究代表者 : 谷口正信 ) によるシンポジウム 計量ファイナンスと時系列解析法の新たな展開 平成 20 年 1 月 24 日 ~26 日香川大学 Realized Volatility の長期記憶性について 1 研究代表者 : 前川功一 ( 広島経済大学 ) 共同研究者 : 得津康義 ( 広島経済大学 ) 河合研一 ( 統計数理研究所リスク解析戦略研究センター ) 永田修一 ( 広島大学大学院社会科学研究科博士課程前期 ) 森本孝之 ( 一橋大学大学院経済学研究科 ) Tee Kian Heng( 岩手県立大学 ) ロク新紅 ( 中国人民銀行金融研究所 ) 主な内容 Part 1 ボラティリティの長期記憶性と GARCH モデル ( 前川 河合 ) Part 2 長期記憶パラメータのウェーブレット推定 ( 永田 得津 前川 ) Part 3 ボラティリティの予測 ( ロク Tee Kian Heng 河合 前川 ) 1 本研究の一部は平成 19~20 年度日本学術振興会科学研究費補助金 ( 基盤研究 (B))( 課題番号 研究課題 高頻度データによる株価 為替レートの計量ファイナンス分析 ( 研究代表者 : 前川功一 )) の補助を受けている
2 Part 1 GARCH () ( ) /24, p.2/93
3 High Frequency Data:, (Ultra High Frequency Data). 1, 5.. :.,. OLSE, NEEDS /24, p.3/93
4 (1),,,. (2),,. (3). (4), /24, p.4/93
5 Volatility 1 e 05 2 e 05 3 e 05 4 e 05 5 e 05 6 e 05 7 e Seconds 1: (7203) :, (2006) /24, p.5/93
6 ARCH/GARCH GARCH(p, q) r t = e t, e t N(0, σt 2 ), e t = σt 2 ɛ t, ɛ t N(0, 1), p q σt 2 = ω + β i σt`1 2 + α j e 2 t`1. i=1 j=1 ω > 0, β i, α j 0(i = 1, 2,, p; j = 1, 2,, q) p q β i + α j < 1 i=1 j= /24, p.6/93
7 GARCH(1,1) GARCH(1,1) r t = e t, e t N(0, σt 2 ), (1) e t = σt 2 ɛ t, ɛ t N(0, 1), (2) σ 2 t = ω + βσ2 t`1 + αe2 t`1. (3) ω > 0, β, α 0, β + α < 1 α + β = 1 integrated GARCH(IGARCH). ω 1 α β /24, p.7/93
8 {y t } h 2 ρ(h) 1 h=0 ρ(h) < ARMA. () 0. 1 ρ(h) = h=0 0. ρ(h) h`, 0 < α < 1 : f(λ) λ ` /24, p.8/93
9 ARFIMA ARFIMA(p,d,q) : Φ(L)(1 L) d y t = Θ(L)ɛ t,, d, {ɛ t } σ 2, Φ(L) = 1 φ 1 L φ 2 L 2 φ p L p, Θ(L) = 1 θ 1 L θ 2 L 2 θ q L q. (1 L) d (1 L) d = k=1 d(d 1)(d 2) (d k + 1) ( L) k k! /24, p.9/93
10 FIGARCH r t GARCH r 2 t ARMA. Φ(L)r 2 t = φ 0 + (1 Θ(L))τ t, τ 2 t = r2 t σ2 t. (1 L) d Φ(L)(1 L) d r 2 t = φ 0 + (1 Θ(L))τ t fractional integrated GARCH(FIGARCH). GARCH α + β, 1 IGARCH FIGARCH, IGARCH, GARCH /24, p.10/93
11 FIEGARCH FIEGARCH(p,d,q) : Φ(L)(1 L) d {log σt 2 ω} = g(z t`1) + γ{ z t`1 E(z t`1 )},, g(z t`1 ) = θz t`1 + γ{ z t`1 E( z t`1 )} z t i.i.d., E(z t ) = 0, V(z t ) = /24, p.11/93
12 (Realized Volatility, RV) GARCH σ t 2.,,. t {r tj, j = 1, 2,, n} 2 RV t RV t = n j=1 r 2 t j (4) /24, p.12/93
13 Volatility Signature Plot.. : (245 ) TOPIX TOPIX 101 RV /24, p.13/93
14 Volatility Signature Plot 800 Average volatility over the sampling period (TOPIX100) Volatility Sampling interval (min.) 2: Volatility Signature Plot : ( ) /24, p.14/93
15 RV GARCH ˆσ 2 t RV t GARCH ˆσ 2 t GARCH(1,1) 225 ˆω = `7, ˆα = , ˆβ = (:9984) ˆω = `7, ˆα = , ˆβ = /24, p.15/93
16 5 2.1a: RV t () GARCH(1,1) ˆσ 2 t () /24, p.16/93
17 5 2.2a: 5 RV t () GARCH(1,1) ˆσ 2 t () /24, p.17/93
18 σ 2 t σ 2 t RV t (1) ( = ) t, j 1 r tj (2) r tj (DGP) r tj = e tj, e tj N(0, σ 2 t j ), (5) e tj = σt 2 j ɛ tj, ɛ tj N(0, 1), (6) σ 2 t j = ω + βσ 2 t j 1 + αe 2 t j 1. (7) : ω = , α = 0.15, β = /24, p.18/93
19 (3) 1 (1440 ) RV 1000 t RV RV t (4) 1 r tj 1 t r t t 1440 σ t 2 σ2 t = j=1 σ tj /24, p.19/93
20 RV t σ 2 t 2.2: RV t () σ 2 t () /24, p.20/93
21 RV t σ 2 t 2.3: RV t () σ 2 t () /24, p.21/93
22 CD-ROM 1 GARCH(1,1) (, ) 5. 2 (,,, ) 6. 3 () /24, p.22/93
23 GARCH(1,1) : r t = e t, e t N(0, σt 2 ), (8) e t = σt 2 ɛ t, ɛ t N(0, 1), (9) σ 2 t = ω + βσ2 t`1 + αe2 t`1. (10) /24, p.23/93
24 1 GARCH(1,1) 3.1: f ˆ ig 1130 i=1 3.2: f ˆ ig 1130 i=1 ˆ = ˆ = X i= X i=1 ˆ i = 0:1522, Var[ ˆ ] = ˆ i = 0:7367, Var[ ˆ ] = X 1130 i= X i=1 ( ˆ i ` ˆ ) 2 = 0:0201 ( ˆ i ` ˆ ) 2 = 0: /24, p.24/93
25 : GARCH(1,1) α(arch ) β(garch ) GARCH(1,1) 1. ˆα, ˆβ 2. Beta(k, l) p, q : f Beta (x) = 1 B(k, l) xk 1 (1 x) l 1, (11) B(k, l) = 1 0 x k 1 (1 x) l 1 dx. (12) /24, p.25/93
26 GARCH(1,1) α, β Beta(k, l) Beta(k, l) : ˆα Beta(ˆk, ˆl ) (13) ˆβ Beta(ˆk, ˆl ) (14) ˆk = , ˆl = ˆk = , ˆl = /24, p.26/93
27 : 3.3: ˆ 3.4: ˆ /24, p.27/93
28 : Beta(p, q) p, q. ˆα Beta(5.4, 30) (15) ˆβ Beta(6, 1.6) (16) 3.5: ˆ 3.6: ˆ /24, p.28/93
29 (1 ) GARCH(1,1) α, β d ((ˆα i + ˆβ i ), ˆd i ). α β 3.7: (( ˆ i + ˆ i; ); ˆd i ) 3.8: f ˆd i g 1130 i=1 E( ˆd) = 0:4301; Var( ˆd) = 0:0977 Skewness( ˆd) = 0:42499 Kurtosis( ˆd) = 2:3987 Median( ˆd) = 0: /24, p.29/93
30 1 ˆd m 1 d ˆd m = : /24, p.30/93
31 (1) ˆα + ˆβ 0.8 ˆd 0. (2) ˆα + ˆβ 0.8 ˆd 0.9. (3) ˆα + ˆβ 1 ˆd 0.5. (4) ˆα + ˆβ 1, ˆd 0.5. =. (5) /24, p.31/93
32 : GARCH(1,1) 1000 GARCH(1,1) 1000 d 4.1: (( ˆ i + ˆ i; ); ˆd i ) 4.2: f ˆd i g 1000 i=1 ˆd = 0:2346; Var( ˆd) = 0:0663 Skewness( ˆd) = 0:5516 Kurtosis( ˆd) = 3:5025 Median( ˆd) = 0: /24, p.32/93
33 α + β 0.9 α Beta(5, 50) β Beta(50, 7) ( E(α) = , E(β) = ) GARCH(1,1) ((ˆα i + ˆβ i, ), ˆd i ). α β 4.3: (( ˆ i + ˆ i; ); ˆd i ) 4.4: f ˆd i g 1000 i= /24, p.33/93
34 ˆd = , Var( ˆd) = Skewness( ˆd) = Kurtosis( ˆd) = Median( ˆd) = /24, p.34/93
35 α + β 0.99 α = 0.09, β = 0.9, 1000 GARCH(1,1) 4.5: α + β = 0.99 ACF /24, p.35/93
36 ˆα + ˆβ 1. GARCH ˆα + ˆβ 1? ˆα + ˆβ 1 GARCH, Granger(1980) /24, p.36/93
37 Part 2 ( ) () () /24, p.37/93
38 DGP: ARFIMA(0,d,0) T=2000, d=0, 0.1(0.1)1.0, = (1) GPH (2) DWT (3) MODWT (4) (F-ML) (5) () (NLSE) (6) /24, p.38/93
39 AFIMA - d = 0 ARMA(p, q) d = 1 ARIMA(p, 1, q) - d 0 < d < ARFIMA(0, d, 0) (1 L) d y t = u t, u t W N(0, σ 2 ) (1) /24, p.39/93
40 (1) GPH log I x (ω k ) = const. ˆd log ( 4 sin 2 ) πω k + εk, (2) ω k = k/t (k = 1,, n < T /2) (2) DWT ln ˆσ 2 (W j ) = const. + (2 ˆd 1) ln τ j. (3) (3) MODWT ln ˆσ 2 ( W j ) = const. + (2 ˆd 1) ln τ j. (4) /24, p.40/93
41 (4) L ( ψ, σ 2) = T log 2π T 2 log σ2 1 2 T j=1 log g (ω j ) 1 2σ 2 I (ω j ) (2) g (ω j ) = f (ω j ) /σ 2. (5) () T j=1 I (ω j ) g (ω j ) (5) T u 2 t = t=1 T t=1 { φ (L) (1 L) d x t θ (L) } 2 (6) (6) l(α) = T log 1 T J w j w j + j=1 h j J T j logh j (7) j= /24, p.41/93
42 ˆd (Discrete wavelet Transform :DWT) Maximum Overlap Discrete Wavelet Transform ( MODWT) /24, p.42/93
43 L 3 n 0 L i=1 h i = 0, L i=1 h 2 i = 1, L i=1 h i h i+2n = 0. (8) L 3 n 0 L i=1 g i = 2, L i=1 g 2 i = 1, L i=1 g i g i+2n = 0. (9) /24, p.43/93
44 g i = ( 1) i`1 h L`i+1 (10) L i=1 g i h i+2n = 0 (11) /24, p.44/93
45 (L = 2) ( 1 h = 2, 1 ) 2 (12) g = ( 1 2, ) 1 2 (13) /24, p.45/93
46 DWT L DWT v 0 x w j;t = v j;t = L h k v (j`1);f(2t`k)mod(n=2 j`1 )g+1, (14) k=1 L k=1 g k v (j`1);f(2t`k)mod(n=2 j`1 )g+1. (15) /24, p.46/93
47 X W = WX (16) T = 16X = (x 0, x 2,..., x 15 ) 0 (L=2) /24, p.47/93
48 1 2 (x 1 x 0 ) 1 2 (x 3 x 2 ) (x 15 x 14 ) 1 2 {x 3 + x 2 (x 1 + x 0 )} 1 2 {x 7 + x 6 (x 5 + x 4 )} 1 2 {x 11 + x 10 (x 9 + x 8 )} 1 2 {x 15 + x 14 (x 13 + x 12 )} {x x 4 (x x 0 )} {x x 12 (x x 8 )} 1 4 {x x 8 (x x 0 )} 1 4 (x x 0 ) = W x 0 x 1 x 2 x 3 x 4 x 5... x 10 x 11 x 12 x 13 x 14 x 15 (17) /24, p.48/93
49 MODWT - DWT j 2 j - MODWT - DWT MODWT h = h/ 2, g = g/ 2. (18) /24, p.49/93
50 MODWT w j ˆσ 2 ( W j ) = N t= L j w 2 j;t N L j + 1 (19) L j (2 j 1)(L 1) /24, p.50/93
51 d d τ j 2 j 1 ˆσ 2 ( W j ) τ j 2d`1 (20) ln ˆσ 2 ( W j ) = const. + (2 ˆd 1) ln τ j (21) MODWT τ j 2 j 1 ˆd /24, p.51/93
52 ARFIMA(0, d, 0) (d 0, 0.1,, 0.6 T = (1) University of Washington Charlie Cornish WMTSA Wavelet Toolkit for MATLAB (2) GAUSS8.0 TSM /24, p.52/93
53 Figure 1: ˆd (d = 0) /24, p.53/93
54 Figure 2: ˆd (d = 0.2) /24, p.54/93
55 Figure 3: ˆd (d = 0.4) /24, p.55/93
56 Table 1: MODWT ˆd Haar D(4) D(8) d ˆd s.d. ˆd s.d. ˆd s.d /24, p.56/93
57 MODWT Table 2: MODWTMLE ˆd MODWT MLE d ˆd s.d. ˆd s.d /24, p.57/93
58 5: 5: 6 ˆd d GPH DWT MODWT F-ML NLSE W-ML /24, p.58/93
59 6: 6: 6 ˆd d GPH DWT MODWT F-ML NLSE W-ML /24, p.59/93
60 : RV ARFIMA d.,, RV ARFIMA(0, d, 0) d /24, p.60/93
61 2 MODWT RV ˆd /24, p.61/93
62 3 MLE RV ˆd /24, p.62/93
63 7: 7: MODWT MLE RV ˆd ˆd MODWT MLE /24, p.63/93
64 4 d = 0, ˆd ( /24, p.64/93
65 5: d = 0, ˆd ( ) /24, p.65/93
66 DWT, MODWT d... DWT, MODWT, W-ML., 1, ARFIMA GARCH /24, p.66/93
67 Part 3 Modeling RV in the Exchange Rate Xinhong Lu () Tee Kian Heng () ( ) () /24, p.67/93
68 Realized Volatility. OLSEN 5 /. : 1991/05/ /08/31, : , 5, Bid Ask., Bid Ask ARFIMAX /24, p.68/93
69 Fig.1 Fig /05/ /08/ /24, p.69/93
70 Fig.2 Fig /05/ /08/ ,10, /24, p.70/93
71 Fig Series: R Sample Observations 3960 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Fig.3., /24, p.71/93
72 Fig.4 Realized Volatility Fig.4 RV. RV = P r t 2. 5 r t : 1/24, 3, 1. p.72/93
73 Fig.5 RV Series: RV Sample Observations 3960 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera 1.16e+08 Probability Fig.5 RV /24, p.73/93
74 Fig.6 log(rv) Series: LN_RV Sample Observations 3960 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Fig.6 log(rv). JB, log(rv) /24, p.74/93
75 GARCH R t GARCH(1,1) : R t = t ; t = ff t z t ; ff t > 0; z t i:i:d:; N(0; 1) fft 2 =! + 2 t`1 + ff2 t`1 ;! > 0; ; 0 GJR,, t 1, 0 D` t,. fft 2 =! + 2 t`1 + ff2 t`1 + D` t 2 t`1 ;! > 0; ; ; 0 (1) t`1 0, D` t = 0, ff 2 t =! + 2 t`1 + ff2 t`1, t`1 < 0, D` t = 1, ff 2 t =! + ( + ) 2 t`1 + ff2 t`1., > 0, /24, p.75/93
76 1: GARCH (a) GARCH : σt 2 = ω + αɛ2 t`1 + βσ2 t`1, ω > 0; α, β 0 t ω ( ) ( ) α ( ) ( ) β ( ) ( ) z-statistic α + β 1, /24, p.76/93
77 Fig.7 GARCH ˆσ 2 t RV Fig.7 GARCH Volatility ˆff 2 t RV /24, p.77/93
78 Fig.8 GARCH ˆσ 2 t RV Fig.8 GARCH Volatility ˆff t 2 RV. ˆff 2 t RV /24, p.78/93
79 RV,, ARFIMAX ( [2006], Andersen et al. [2003] ).,,, Giot and Laurent[2004], AIC ARFIMAX(p, d, q) p, q 1,, ARFIMAX(0,d,1). ARFIMAX(0,d,1): AIC = ARFIMAX(1,d,1): AIC = ARFIMAX(1,d,0): AIC = /24, p.79/93
80 RV-ARFIMAX(0,d,1) n o (1 ` L) d ln(rv t ) ` 0 ` 1 jr t`1 j ` 2 D` t`1 jr t`1j D` t`1 = = (1 + L)u t ; u t i:i:d:n(0; ffu). 2, (d; 0 ; 1 ; 2 ; ), D` 8 < : (2) t`1 0 R t`1 0 1 R t`1 < 0, ln(rv t ) R t`1, E[ln(RV t )jr t`1 ] = 8 < : jr t`1 j R t` ( )jr t`1 j R t`1 < 0. L, (1 ` L) d. (1 ` L) d = 1 + 1X k=1 d(d ` 1) (d ` k + 1) (`L) k k! 0 < d < 1,, d < 0:5, d 0: /24, p.80/93
81 ,. u t., t-1 t RV-ARFIMAX(0,d,1) RV RV tjt`1.» drv tjt`1 = exp 0 + ( D` t`1 )jr t`1j` 1X k=1 d(d ` 1) (d ` k + 1) (`1) k k! + û t` ˆff2 u n o ln(rv t`k ) ` 0 ` ( D` t`1 )jr t`k`1j (3), û t (2), ˆσ 2 u., (3), RV tjt`1, /24, p.81/93
82 Fig.12 Rolling Window Fig.12 Rolling Window (windowsize=500 ). 1, /24, p.82/93
83 2: 2 : (windowsize = 500, rolling window) ˆd ˆ 0 ˆ 1 ˆ 2 ˆff u 2 Failure Rate % % % % % % % 2. 0 < d < 1,. d < 0:5, /24, p.83/93
84 3: ˆd 3 : ˆd ˆd ,,,, ˆd., ˆd /24, p.84/93
85 Fig ˆd Fig.13, Rolling Window RV-ARFIMAX ˆd ˆd. Fig.14 1, Rolling Window RV-ARFIMAX ˆd ˆd /24, p.85/93
86 Fig.15 RV RV Fig.15 RV-ARFIMAX Volatility d RV RV /24, p.86/93
87 Fig.16 RV RV Fig.16 RV-ARFIMAX Volatility d RV RV /24, p.87/93
88 Fig.17 RV, RV med Fig.17 RV-ARFIMAX Volatility d RV med RV /24, p.88/93
89 Fig Fig.18 RV-ARFIMAX Volatility d RV RV d RV med 5%VaR-Threshold /24, p.89/93
90 ,, RMSE (root mean squared error), RMSPE (root mean squared parcentage error), MAE (mean absolute error), MAPE (mean absolute parcentage error). RMSE = q 1 n P n t=1 (RV t ` ˆff 2 tjt`1 )2 RMSPE = r 1 n P n t=1 ( RV t`ˆff 2 t t 1 RV t ) 2 MAE = 1 n P nt=1 jrv t ` ˆff 2 tjt`1 j MAPE = 1 n P n t=1 RV t`ˆff 2 t t 1 RV t, ˆff 2 tjt`1 t-1 t ff2 t /24, p.90/93
91 4: () 4 : () RMSE RMSPE MAE MAPE RV-ARFIMAX-mean RV-ARFIMAX-med GARCH-n GARCH-t GJR-n GJR-t EGARCH-n EGARCH-t APGARCH-n APGARCH-t RMSPE, RV-ARFIMAX /24, p.91/93
92 - 1 OLSEN /Realized VolatilityRV,RV.. (1) Hurst R/S R/S. (2), (1 ` L) d d. (3) ARFIMAX(p,d,q). (4) RV GARCH, RV. RV. (5) GARCH ARFIMAX(0,, ) RV RMSE. ARFIMAX(0,, ). RV /24, p.92/93
93 - 2,.,, d, , /24, p.93/93
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