低金利環境下でのフィット向上を目指した最近のイールド･カーブ･モデル群

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1 IMES DISCUSSION PAPER SERIES 低金利環境下でのフィット向上を目指した最近のイールドカーブモデル群ふじわらしげあき藤原茂章 Discussio Paper No. 7-J- INSIUE FOR MONEARY AND ECONOMIC SUDIES BANK OF JAPAN 日本銀行金融研究所日本橋郵便局私書箱 3 号日本銀行金融研究所が刊行している論文等はホームページからダウンロードできます hp:// 無断での転載複製はご遠慮下さい

3 IMES Discussio Paper Series 7-J- 7 低金利環境下でのフィット向上を目指した最近のイールドカーブモデル群ふじわらしげ藤原茂章あき * Blac[995] JEL classificaio: E43 G * shigeai.fuiwara@bo.or.p

4 Vasice[977] Rogers[996] % Blac[995] shaow ieres Gorovoi a Liesy[3, 4] Miller a Plae[4] Bomfim[3]

5 Blac[995] Gorovoi a Liesy[3,4] Blac[995] Miller a Plae[4] GOP Bomfim[3] [3] [5] Kabaov, Kiima a Riaz[5] Blac[995] CIR Cox, Igersoll a Ross[985] 93 γ r θ r + σr W γ,. 5 CIR γ [999]

6 CIR 3 CIR X r + r X max[ X,], CIR θ σ Feller[95] 4 CIR X θ X + σ X + lw, l > 5 3

7 Gorovoi a Liesy[3,4] X 6 X θ X + σw, X x, τ P x, τ τ + P x, τ E[exp X u u], 3 P x, τ P x, fuameal parial iffereial equaio 7 σ + Pxx + θ x Px x P Pτ. 4 Px P x x + x + x max[x,] 4 P x, τ ce τ ϕ x, 5 { } < < <... lim { ϕ } e τ τ ϕ c 6 CIR 7 4

8 5 3,,, σ θ γ θ σ α σ c c. 6 ] [ + α α α γ α γ γ γ D D cd D D c w 7 z D v parabolic cylier fucio z H e z D v z v v, 8 z H v v z v z v v z e e z H, 9 +, ', ' 4 4 x if cx D e D w D x if cx D e D w D x cx cx α α α ϕ γ γ γ. w w / ' x e x c x σ θ σ ϕ,, τ x R

9 l P x, τ lim R x, τ lim >, τ τ τ θ,, σ x 3 4bp Liesy[4] [6] firs hiig ime of zero GOP Miller a Plae[4] Growh Opimal Porfolio; 6

10 GOP 8 GOP 9 GOP B B B r, 3 r primary securiies X,...,, X X { a + b W },,...,, 4 a [ a ], b [ b ], 8 self-fiacig GOP Plae[] 9 GOP GOP ArrowDebreu 7

11 η η, η,..., η b [ a r ], 5 i δ S i S S r + r, δ X b η + W, 6 X S GOP η 6 δ X b, S S { r + η η + W }, 7 GOP GOP S 8 S S { r + η η + W }. 8 η W η GOP W GOP MSCI covariaio W W 8

12 η GOP GOP S S S, 9 B GOP S Sη η + W, GOP r W GOP P, GOP S B P, S E A E A, S S B A GOP P, P, M, S G, r. GOP 9

13 S M, S E A, 3 S B G, r E A, 4 B GOP GOP S α α S η, 5,5 GOP S α + α S W. 6 ξ Plae[4] ξ α ss. 7 4 Y ξ S, 8 6 Y ξ ξ Y ξ Wξ, 9 W α W 4 ξ, 3 9 Y ξ 4 square Bessel

14 process Revuz a Yor[999] S 4 S < S > α ss, 3 4 GOP α α exp{ ω}, 3 α, ω 3,6 4 M, S ωs exp{ }. 33 α α M

15 α., ω.5, η.. M M r X + r X max[ X,], 34 G, X Eexp{ max X u, u} A, P, 33 M, S

16 P, G, X, 36 M, S G, X CIR CIR Bomfim[3] X X X X θ X σ W θ θ θ θ θ σ W, 38 θ α + X θ W, W X, θ 3

17 LIBOR [3] 3 τ ψ Ψ r * { τ } { µ r + σ W }, 39 θ 3 [5] [6] 5 CIR 4

18 P, P ψ s H Ψ, Ψ s s +, Ψ H µ σ / σ H H exp{ }, 4 4 H exp [5] h * * * * * * P, H s h sexp Φ + Φ s + exp Φ + Φ, s H µ σ / σ H H exp{ }, 4 4 H exp 5

19 Φ * * h u u Kabaov, Kiima a Riaz[5] 4 r { r r + W + r J r r > } [ θ σ ] { }, Z J { r Z Z Z } [ ],. 4 J r Z W J Z laecy 4 6

20 4 r Z Z J Z r Z h z α + γz γ P r, Z, P r, Z, E[exp rs s r r, Z Z ] parial iegro-iffereial equaio r, Z S S Z S Z 5 5 Z Z Z 7

21 6 Gorovoi a Liesy[3,4] 7 Miller a Plae[4] Miller a Plae[4] GOP GOP 6 Blac[995] Gorovoi a Liesy[3] I-S 7 CIR 8

22 GOP 9

23 GOP GOP W b S X b r a S X W b S X b r a S X S X,,,, } { } { η η η η η η, A- ] [ r a b η r a b, η A- W b S X S X, η, A- GOP GOP

24 Blac, F., Ieres as opios, Joural of Fiace, 57, 995, pp Bomfim, A., Ieres rae as opios: Assessig he mares view of he liquiiy rap, Fiace a Ecoomics Discussio Series, he Feeral Reserve Boar, No. 3-45, 3. Cox, J. C., J. E. Igersoll, a S. A. Ross, A heory of he erm Srucure of Ieres Raes, Ecoomerica, 53, 985, pp Feller, W., wo sigular iffusio problems, Aals of Mahemaics, 54, 95, pp Gorovoi, V. a V. Liesy, Shaow Ieres, Ris, 6, 3, pp Gorovoi, V. a V. Liesy, Blac s Moel of Ieres Raes as Opios, Eigefucio Expasios a Japaese Ieres Raes, Mahemaical Fiace, 4, 4, pp Kabaov, Y., Kiima, M., a S. Riaz, A Posiive Ieres Rae Moel wih Sicy Barrier, Discussio Paper No.7, Kyoo Uiversiy, 5. Liesy, V., he Specral Decomposiio of he Opio Value, Ieraioal Joural of heoreical a Applie Fiace, 73, 4, pp Liesy, V., Compuig hiig ime esiies for CIR a OU iffusios: applicaios o mea-reverig moels, Joural of Compuaioal Fiace, 74, 4, pp.- Log, J. B., he umeraire porfolio, Joural of Fiacial Ecoomics, 6, 99, pp Miller, S. a E. Plae, A wo-facor Moel for Low Ieres Rae Regimes, Asia-Pacific Fiacial Mares,, 5, pp Plae, E., Arbirage i coiuous complee mares, Avaces i Applie Probabiliy, 343,, pp Plae, E., Moelig he volailiy a expece value of a iversifie worl iex,

25 Ieraioal Joural of heoreical a Applie Fiace, 74, 4, pp Revuz, D. a M. Yor, Coiuous Marigales a Browia Moio, Spriger, 999. Rogers, L., Gaussia errors, Ris, 9, 996, pp Vasice, O., A Equilibrium Characerizaio of he erm Srucure, Joural of Fiacial Ecoomics, 5, 977, pp , B. L. Merrill Lych Fixe Icome Sraegy J- 6 3-J

デフォルト相関係数のインプライド推計（）

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