II III II 1 III ( ) [2] [3] [1] 1 1:
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1 II III II 1 III () [2] [3] [1] 1 1:
2 [5] [6] () [7] [1] [1] [1] [1] t Dt L DF t A t (2.1) A t = Dt L + Dt F (2.1) QQ Shapiro-Wilk 3 p () QQ p () QQ 1 1 R ecompose 2 1 µ 3 Shapiro-Wilk 2
3 h Œ µ µ Ø Ø f { Ø Ø Ø 2: 3: QQ 3
4 4: QQ A t λ t (2.3) ( ) At+1 λ t = ln (2.2) A t λ t = µt + σz (2.3) () λ t 1 [1] 1: µ σ µ σ [1] µ σ µ 2 λ t (DW) DW 2.819(p : 1 : ) DW 2.199(p : ) p 0.1% p 10% [1] p p 10% λ t 4
5 () 5 l l l l e e Œ>/KZ el el l l e e Œ >/KZ el el l l l l l l 5: 1 LIBOR 1% e e e e eee e ee e e e e e eee e e ee e e e 6:
6 (2.4) ρ t = DF t D L t (2.4) D L t t DF t t 7 e e e eee e ee e eee e ee e 7: [1] e e e Œ >/KZ l l el el l l Œ >/KZ l l el el l l l l l l 8: 6
7 [1] [4] (2 ) ( 9) l 9: µpµµ µµ µµµ µ µ µ µµ µ xœµ µµµ µµ µµµµ µµ µµµ µµµµ oµµµ µ µµ LIBOR :
8 r t ln r t : ρ t = α 1 (r t α 2 )t + α 3 (2.5) 2 : ρ t = α 1 (r t α 2 )t + α 3 r t + α 4 (2.6) 3 : ρ t = (α 1 ln r t α 2 )t + α 3 (2.7) 4 : ρ t = (α 1 ln r t α 2 )t + α 3 ln r t + α 4 (2.8) ( ) : 1 α 1 ( 10 3 ) α 2 ( 10 3 ) α 3 ( 10 2 ) α 4 ( 10 2 ) α 1 ( 10 3 ) α 2 α 3 α α 1 ( 10 3 ) α 2 ( 10 3 ) α 3 ( 10 2 ) α 4 ( 10 2 ) α 1 ( 10 3 ) α 2 ( 10 2 ) α 3 α α 1 ( 10 3 ) α 2 ( 10 3 ) α 3 ( 10 2 ) α 4 ( 10 2 ) α 1 ( 10 3 ) α 2 ( 10 2 ) α 3 α α 1 ( 10 3 ) α 2 ( 10 3 ) α 3 ( 10 2 ) α 4 ( 10 2 ) α 1 ( 10 3 ) α 2 ( 10 2 ) α 3 α : *** 0.1%, ** 1%, * 5% α 1 (r t α 2 )t = (α 1 ln r t α 2)t, α 2 = α 1 α 2 α 2 p α 2 α 2 = α 2/α 1 [1] 8
9 [1] 6 ( ) [1] 1 α % % 4 α 1 α 3 [1] α µµ Œ l µµ Œ l 11: ( 1) µµ Œ l µµ Œ l 12: ( 2) 9
10 µµ Œ l µµ Œ l 13: ( 3) µµ Œ l µµ Œ l 14: ( 4) [1] [1] [1]
11 µ } µ } µµ µµ 15: [1] [1] 2 4 [1] [1] 3 0.1% 50,000 1 (2.3) (3.1) λ t (2.2) (3.2) A t+1 ε t λ t = λ t 1 + λ t, λ t = µ t + σε t t (3.1) A t+1 = A t e λ t (3.2) 2 (3.4) Hull-White ρ t r t = (θ t ar t )t + σz t (3.3) ρ t = (α 1 ln r t α 2 )t + α 3 ln r t + α 4 + σ e e t (3.4) σ e e t 11
12 ε t e t Hull-White a = , σ = A t ρ t D L t = A t 1 + ρ t (3.5) 4 E(D L t ) 99% Volume at Risk VaR(D L t ) a. X e 0 = D L 0 E(D L 1 ) X e t = E(D L t ) E(D L t+1), (t = 1,..., T 1) X e T = E(D L T ) b. Volume at Risk(VaR) X v 0 = D L 0 VaR(D L 1 ) X v t = VaR(D L t ) VaR(D L t+1), (t = 1,..., T 1) X v T = VaR(D L T ) Hull-White θ t 16() 2 A : B : e Ø Ø e e e e e 16: A 16 6 ln(r t )t α 1 ln(r t ) α 3 α 1 < α 3 t 6 B A A 12
13 α 3 ln r t (α 1 t α 3 ) ln r t t α 1 t 3.2. A B 17 17: 10 α 1, α 3 B A B ( ) B : (, A) 13
14 : (, A) : (, B) : (, B) 14
15 3.4. () X t (3.6) T z L = T Xt z t t=1 D0 L (z {e, v}) (3.6) e v VaR VaR 3 3: () A VaR VaR VaR B VaR VaR VaR A VaR AA-Kijima [2] ( ) AA-Kijima 4 AA-Kijima 4 AA-Kijima () β Y z 0 = X z 0 + T t=1 Y z t = (1 β)x z t 15 βx z t
16 4: AA-Kijima AA-Kijima(*1) µ % 9.5% 12.4% (2007) σ 4.9% 4.0% 4.1% () () () () () () (*2) () () () (2011) () () () () () () (*1) ( [2]) (*2) (3.7) T T Xt z = Yt z = D0 L (z {e, v}) (3.7) t=0 t=0 T z C = T Yt z t=1 D0 L t = (1 β)t z L (z {e, v}) (3.8) 30%(β = 0.3) 5 5: (: 30%) () A VaR VaR VaR B VaR VaR VaR
17 (2.8) : [1] 7: 4 α 1 ( 10 3 ) α 2 ( 10 3 ) α 3 ( 10 2 ) α 4 ( 10 2 ) α 1 ( 10 3 ) α 2 ( 10 2 ) α 3 α : *** 0.1%, ** 1%, * 5% 7 α 1 α 3 2 α 1 α 3 α 1 α 3 0.1%
18 e e e e e e e e e e e 22: () e e e e e e e e e e e e e e e e e e e e e 23: () e e 18
19 5. [1] ( ) [1], :,, , pp [2] :, BOJ Reports & Research Papers, [3], [4],, [5],,,, 2012, pp [6],, SAS All Analytics [7],,,,,,,,, ( 9, pp ), ,. 19
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