「国債の金利推定モデルに関する研究会」報告書

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1 : LG LG Quadratic Gaussian ,,, kijima@center.tmu.ac.jp, tanaka-keiichi@tmu.ac.jp 1

2 L G 2 1 L G r L t),r G t) L r L t) G r G t) r L t) h G t) =r G t) r L t) r L t) MMC; money market account R t B L t) =e 0 r Ls)ds Q L Ω, F,Q L ) Q L 2 W L t),w G t)) {F t } 2 X P E P X 2.1 L L P L t, T ) P L t, T )=E Q L e R T r t L s)ds F t = E Q BL t) L F t 1) B L T ) P L t, T ) Q T L dq T L dq L Ft = P L t, T ) B L t)p L 0,T) 2) t T 1,T 2 ) 1 PL t, T 1 ) T 2 T 1 P L t, T 2 ) 1 LIBOR LIBOR t T 0,T N St, T 0,T N ) t T 0 <T 1 < <T = T N δ i = T i T i 1 t T i 1,T i LIBOR Lt, T i 1,T i ) P L t, T ) Lt, T i 1,T i )= 1 E QT i 1 L δ i P L T i 1,T i ) 1 Ft 1 L, G LIBOR Government Bond ) 2

3 St, T 0,T N ) LIBOR LT i 1,T i 1,T i ) i =1, 2,...,N) P L t, T 0 )+St, T 0,T N ) = P L t, T 0 )+ δ i P L t, T i )+P L t, T N ) δ i Lt, T i 1,T i )P L t, T i )+P L t, T N ) 3) St, T 0,T N )= N δ ilt, T i 1,T i )P L t, T i ) N δ ip L t, T i ) = P Lt, T 0 ) P L t, T N ) N δ ip L t, T i ) 4) δ i Lt, T i 1,T i )P L t, T i ) = = E QT i L 1 P L T i 1,T i ) 1 Ft P L t, T i ) PL t, T i 1 ) P L t, T i ) = P L t, T 0 ) P L t, T N ) ) 1 P L t, T i ) P L t, T ) L 2.2 L L r G t) r L t) r G t) G r G t) MMC B G t) =e R t 0 r Gs)ds 5) λ L t), λ G t) B G t) Q G dq G Ft = ζ G t) dq L { t exp 0 λ L s)dw L s) λ G s)dw G s) 1 λ L s) 2 + λ G s) 2) )} ds 2 6) 3

4 6) G P G t, T )=E Q BG t) G F t 7) B G T ) 3 G r G t) r L t) h G t) =r G t) r L t) 4 L LIBOR G t T 1,T 2 ) 1 PG t, T 1 ) T 2 T 1 P G t, T 2 ) 1 Govt L Govt t Govt Gt, T i 1,T i ) Gt, T i 1,T i )= 1 E QT i 1 ) L F t 1 8) δ i P G T i 1,T i ) T N CT N ) V t, T N ) Gt, T i 1,T i ) V t, T N ) V t, T N )+CT N ) = P L t, T 0 )+ V t, T N )=P L t, T 0 )+ δ i P L t, T i )+P L t, T N ) δ i Gt, T i 1,T i )P L t, T i )+P L t, T N ) 9) δ i CT N ) Gt, T i 1,T i ))P L t, T i ) 10) N δ igt 0,T i 1,T i )P L T 0,T i ) N δ ip L T 0,T i ) 3 P Gt, T ) L 4 h Gt) 4

5 9) Govt GT i 1,T i 1,T i ) 9) L G = CT N ) δ i P L t, T i ) = P L t, T 0 ) δ i Gt, T i 1,T i )P L t, T i ) Lt, T i 1,T i ) Gt, T i 1,T i ) Govt LIBOR Gt, T i 1,T i ) <Lt, T i 1,T i ), i =1, 2,...,N V t, T N ) >CT N ) δ i P D t, T i )+P D t, T N ) 11) L 3 LG r L t) Q L Quadratic Gaussian r L t) = yt)+α + βt) 2, dyt) = a L yt)dt + σ L dw L t), y0) = y 0 12) h G t) dh G t) =θt)dt + σ G dw G t), h G 0) = h 0 θt) Nelson-Siegel ft) ft) = β 0 + β 1 exp t ) t + β 2 exp t ) 13) τ 1 τ 1 τ 1 θt) = dft) + σg 2 dt t 14) 5

6 λ L t) =0, λ G t) =λ G 15) G B G t) Q G dq G Ft = ζ G t) = exp 1 ) dq L 2 λ2 Gt λ G W G t) Quadratic Gaussian L P L t, T ) P L t, T ) = exp A L t, T ) B L t, T )yt) C L t, T )yt) 2) 16) Pelsser 1997) A L t, T ),B L t, T ), C L t, T ) γ = a 2 L +2σ2 L, F L t, T ) = ) 1 2γe γt t) γ + a L )e 2γT t) + γ a L, C L t, T ) ) ) 1 = e 2γT t) 1 γ + a L )e 2γT t) + γ a L, T α + βs B L t, T ) = 2F L t, T ) t F L s, T ) ds = 2B 1 γ 2, A 5 T ) 1 A L t, T ) = t 2 σ2 L B Ls, t) 2 σl 2 C Ls, T ) α + βs) 2 ds ) = σl 2 A4 γ 5 + A 6 α 2 T t) αβt 2 t 2 ) 1 A 5 3 β2 T 3 t 3 ), Γ 1 = γ a L, Γ 2 = γ + a L, A 1a = e γt t) +4 e γt t) 3 + 2γT t)), A 1b = e γt t) 4+e γt t) 3 2γT t)), A 2a = e γt t) 1 γt) 21 γt + T )) + e γt t) 1 γ2t + T )+γ 2 t 2 T 2 )), A 2b = e γt t) 1 + γt) 21 + γt + T )) + e γt t) 1 + γ2t + T )+γ 2 t 2 T 2 )), A 3a = 4γt1 γt) e γt t) 1 γt) 2 + e 1+2γt γt t) γ 2 2t 2 + T 2 )+ 2 ) 3 γ3 t 3 T 3 ), A 3b = 4γt1 + γt)+e γt t) 1 + γt) 2 + e 1+2γt γt t) + γ 2 2t 2 + T 2 )+ 2 ) 3 γ3 t 3 T 3 ), A 4 = α 2 γ 2 Γ 1 A 1a +Γ 2 A 1b )+2αβγΓ 1 A 2a +Γ 2 A 2b )+β 2 Γ 1 A 3a +Γ 2 A 3b ), A 5 = Γ 1 e γt t) +Γ 2 e γt t), A 6 = 1 2 T t) Γ 1 1 Γ 1 ) Γ 1 1 +Γ 1 ) 2 ln A5 ln2γ)), 2γ B 1 = αγ e γt e γt) Γ 1 e γt +Γ 2 e γt) ) + β Γ 1 e γt t) 1 γt)+γ 2 e γt t) 1 + γt) Γ 1 1 γt) Γ γt) 6

7 G P G t, T ) P G t, T ) = E Q BG t) G F t B G T ) = E Q BG t) ζ L G T ) F t B G T ) ζ G t) = E Q L Bt) ζ G T ) R T BT ) ζ G t) e h t G s)ds F t = E Q BL t) L F t E Q ζg T ) R T L B L T ) ζ G t) e t h G s)ds F t = P L t, T )E Q G e R T t h G s)ds F t 17) ζ G t),h G t) r L t) 17) H G t, T )=P G t, T )/P L t, T ) H G t, T )=E Q G e R T t h G s)ds F t 18) h G t) H G t, T ) G Q G Q G ht) dh G t) =θt) σ G λ G )dt + σ G dw G G t) WG Gt) =W Gt)+λ G t Q G H G t, T ) 5 H G t, T ) = exp {A G t, T )+B G t, T )h G t)} 19) B G t, T ) = T t), A G t, T ) = = T t T t θs) σ G λ G )T s)ds + σ2 G 2 T fs)ds +T t)ft) σ2 G T t)2 2 t T s) 2 ds t λ ) G σ G 5 Hull White 1990) 7

8 Govt δ i Gt, T 1,T 2 ) = E QT 2 1 L F t 1 P G T 1,T 2 ) = E QT 2 PL T L 1,T 1 ) P L T 1,T 2 ) e A GT 1,T 2 ) B G T 1,T 2 )h G T 1 ) F t 1 = P Lt, T 1 ) T 1 P L t, T 2 ) EQ L e A GT 1,T 2 ) B G T 1,T 2 )h G T 1 ) F t 1 = P Lt, T 1 ) P L t, T 2 ) EQ L e A GT 1,T 2 ) B G T 1,T 2 )h G T 1 ) F t 1 = P Lt, T 1 ) { P L t, T 2 ) exp A G T 1,T 2 ) B G T 1,T 2 )E Q L h G T 1 ) F t + 1 } 2 B GT 1,T 2 ) 2 Var Q L h G T 1 ) F t 1 3 T 2 T 1 4 h G t) W L t) 5 h G T 1 ) F t h G t) T1 E Q L h G T 1 ) F t = h G t)+ θs)ds t Var Q L h G T 1 ) F t = σg 2 T 1 t) Govt Gt, T 1,T 2 ) = 1 ) PL t, T 1 ) δ i P L t, T 2 ) K Gt, T 1,T 2 ) 1 20) K G t, T 1,T 2 ) { = exp A G T 1,T 2 ) B G T 1,T 2 ) { T 2 = exp fs)ds +T 2 T 1 )h G t) ft)) T 1 + σ GT 2 T 1 ) 2 T1 ) h G t)+ θs)ds + σ2 G T 2 T 1 ) 2 } T 1 t) t 2 } T 2 T 1 )σ G T 1 λ G )+σ G T 1 t)t 2 + t)) P L t, T 1 )/P L t, T 2 ) LIBOR K G t, T 1,T 2 ) LIBOR Govt K G t, T 1,T 2 ) < 1 Govt LIBOR 21) 8

9 V t, T N ) = P L t, T 0 )+ = P L t, T 0 )+ = δ i CT N ) Gt, T i 1,T i ))P L t, T i ) 1 + δ i CT N ))P L t, T i ) K G t, T i 1,T i )P L t, T i 1 )) δ i CT N )P L t, T i )+P L t, T N )+ 1 K G t, T i 1,T i )) P L t, T i 1 ) 22) 3 N 1 K Gt, T i 1,T i )) P L t, T i 1 ) L 2 ZV t, T N ) ZV t, T N ) = P L t, T N )+ 1 K G t, T i 1,T i )) P L t, T i 1 ) 23) 4 30 LG 3 L r L t) a L, σ L, y 0, α, β G h G t) σ G, h 0, λ G, β 0, β 1, β 2, τ 1 r L t) LIBOR 1. min n S model T i ) S obs T i )) 2 S model T i )=S0,T 0,T i )= P L0,T 0 ) P L 0,T i ) i k=1 δ kp L 0,T k ) S obs T i )= T i LIBOR 9

10 2. LIBOR:1m,3m,6m LIBOR: 1-10yr,15,20,25,30yr Source : British Bankers Association, Bloomberg) 3. Nelder-Mead 6 Excel OPTMIZ v : r L t) a L σ L y α β h G t) r L t) 1. 8 min n Z model T i ) Z obs T i ) Z model T i )= ln P L 0,T i )+ i / 1 K G 0,T k 1,T k )) P L 0,T k 1 )) T i T 0 ) k=1 Z obs T i )= ln V obs T i ) CT i ) i / δ k P L 0,T k )) T i T 0 ) k=1 V obs T i ) = JGB T i CT i ) Source : ) 3. Nelder-Mead Excel OPTMIZ v Nelder-Mead Nelder and Mead 1965) 7 Xnumbers5.4 8 robust 10

11 2: h G t) σ G h λ G β β β τ : JX26-7 JX26 JX25 JX24 JX /7/ /4/ /1/ /10/ /7/12 a L σ L y α β σ G h λ G β β β τ JX22 JX21 JX20 JX19 JX /4/ /1/ /10/ /7/ /4/13 a L σ L y α β σ G h λ G β β β τ

12 JX17 JX16 JX15 JX /2/ /10/ /8/ /4/14 a L σ L y α β σ G h λ G β β β τ JGB V h G t) Nelson- Siegel Quadratic Gaussian 12

13 4: %) 30 %) 40 %) bp) JX / JX / JX / JX / JX / JX / JX / JX / JX / JX / JX / JX / JX / JX / Hull-White Quadratic Gaussian Vasicek CIR 1 Nelson-Siegel , 2007),,. 2 Hull, J., and A. White. 1990). Pricing Interest-Rate-Derivative Securities. Review of Financial Studies, 3, Nelder, J.A., and R. Mead. 1965). A simplex method for function minimization. The Computer Journal, 7, Pelsser, A. 1997). A Tractable Yield-Curve Model That Guarantees Positive Interest Rates. Review of Derivatives Research, 1,

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