,,,.,.,.,,.,,.,,.,,. :,, OU, SUR,.,.,,.,.., 62
1, BIS,.,.,,,,.,,.,.,, [33],,,.,,.,,,,.,,.,,,.. 2,. 3,. 4,,. 5. 2,,.,. 2.1,.,,.,. Giesecke [10] dependen defaul.,,,, GDP,.,,.,...,,,.,.,.,,.,,. Giesecke [10], 1. 63
1 1-(1) (,, ) 1-(2) 1-(3) 2 ( ) 2-(1) 2-(2) 2-(3) 1:.,, 2. ( ) ( ) Meron [22] Duffie and Singleon [7] Jarrow and Turnbull [15] 2:.,..,,..,,. 2.2 2..,,. 2.,,,.,.,.,. 2,., Meron [22].. 64
,.,.,,.,.,,, (Gordy [13])., JP Morgan [16] CrediMerics T M,., Jarrow-Turnbull [15], Duffie-Singleon [7].., 1.,,.,..,.,,.,,.,.,,.,. Kusuoka [21], Jeanblanc and Rukowski [25], Collin-Dufresne, Goldsein and Hugonnier [4].,, (Duffie, Pan and Singleon [6]).,, OU. OU, Aonuma and Nakagawa [1],, [29].. 2.3, (Ω, F, P).,, F, (0 )., (0 < ).,. 1 {A} : A 1, 0, : (0 < ), T : ( T < ), X(, T ) : T, X 0 (, T ) : T, τ : (0 τ < ), δ :, L() :, r 0 () :. 1.,,. 65
N() = 1 {τ },. lim s N() = N( ).,. G : h. H : N(s) (0 s T ).,.,. F., F G H. X(, T ), {τ T }. τ.,. Duffie and Singleon [9].., 1,.,, P - h(). P[ < τ + F ] 1 {τ>} h() = lim 0 (1), > 0., h(). h(), h() 0 h(s)ds = Y () E [exp { H(, T )} F, τ > ] 1. 2 H(, T ) = T h(s)ds. H(, T ) T., P[τ > T F ] = 1 P[τ T F ], (Duffie[5]). [ { ] T P[τ > T F ] = 1 {τ>} E exp h(s)ds} G (2),., P, r 0 (). X 0 (, T ), r 0 (). [ { ] T X 0 (, T ) = Ẽ exp r 0 (s)ds} G (3), r 0 () 0, r 0 (s)ds =. (3) (2),, {τ 0 > }, {τ 0 > T } T 1, {τ 0 T }. τ 0, r 0 3. h 0 {τ 0 > T }, T P h 0 () = r 0 () (0 T ). H 0 (, T ) = T h 0 (s)ds.,., δ τ,., P.,. 2, Y () Y () lim s Y (s), Y () = 0 a.s. 3 ([32]). 66
X(, T ) = 1 {τ>} Ẽ [ 1 {τ>t } exp { H 0 (, T )} + 1 {τ T } δ τ exp { H 0 (, τ)} F ]. (4),. Duffie and Singleon [7], 1 δ, 3. RMV(recovery of marke value) δ = (1 L())X(, T ) RT(recovery of reasury) δ = (1 L())X 0 (, T ) RFV(recovery of face value) δ = (1 L()) L(). RMV, X(, T ) T,,. Duffie and Singleon [7]., Jarrow and Turnbull [15] RT., Jarrow and Turnbull [15]. (4) RT, L() = L, δ 1 L,,. { } X(, T ) = 1 {τ>} X 0 (, T ) δ + (1 δ)ẽ [exp { H(, T )} G ] (5) A., (, T ),. 3 (, T ) = 1 ( ) X(, T ) T log X 0 (, T ) = 1 ( ) T log δ + (1 δ)ẽ [exp { H(, T )} G ],,.,. 3.1,,..,,.,.,.,.,.,,. 3. (6) 3.2 1,. 1,, n. i (1 i n), h i (), τ i. {τ i T }.,., T,, 67
A. B (h A ()) (h B ()) ( ) ( ) ({τ A T }) ({τ B T }) ( ) ( ),. 3:. A B 2.,..,.,. {τ i > } (i = 1, 2,..., n), [ { ] n P[τ 1 > 1,, τ n > n F ] = E exp H i (, i )} G. (7) (7) A. Kijima [18], Kijima and Muromachi [19]., Kijima and Muromachi [20]. 3.3,. {τ i > } (i = 1, 2,..., n)., T i j (i j) 1 {τi T }, 1 {τj T }., 4. (7), ρ(1 {τi T }, 1 {τj T } F ). 1 {τi T } 1 {τj T }. i=1 Cov(1 {τi T }, 1 {τj T } F ) = E[1 {τi T }1 {τj T } F ] E[1 {τi T } F ] E[1 {τj T } F ] = P[τ i T, τ j T F ] P[τ i T F ] P[τ j T F ] (8), (7),. 4 Giesecke [10].,,,,,.,. 68
1: A B. (9), A B. ρ(1 {τa T }, 1 {τb T } F ) = {(1) ((1) + (2))((1) + (3))}/{((1) + (2))((3) + (4))((1) + (3))((2) + (4))} 1/2. (1) + (2) + (3) + (4) = 1. B A A P[τ A T, τ B T F ] (1) P[τ A T, τ B > T F ] (2) P[τ A T F ] (1)+(2) P[τ A > T, τ B T F ] (3) P[τ A > T, τ B > T F ] (4) P[τ A > T F ] (3)+(4) B P[τ B T F ] (1)+(3) P[τ B > T F ] (2)+(4) 1 (1)+(2)+(3)+(4) ρ(1 {τi T }, 1 {τj T } F ) = Cov(1 {τi T }, 1 {τj T } F ), (i j) (9) V ar(1 {τi T } F )V ar(1 {τj T } F ) 1., ρ(1 {τi T }, 1 {τj T } F ) = ρ(1 {τi>t }, 1 {τj>t } F ). = T,, 0. 4 Ornsein-Uhlenbeck,. OU,.,,,.,. 4.1 OU Vasicek,., OU,. h() = (h 1 (), h 2 (), h n ()) OU. 5 dh i () = (a i b i h i ())d + σ i dw i (), 0. (10), a i /b i = h i,. dh i () = b i (h i h i ())d + σ i dw i (), 0. (11), P h i h i (), b h i, σ i. W (), ρ. ρ ii = 1. ρ ji. dw i ()dw j () = ρ ij d (12) 5 OU,, h().,. OU., CIR. dh = b(h h())d + σh() c dw (), 0. h(0) 0, c = 1,, 2.. 69
β i () i, T W i () = W i () + P. β i (), φ i (). 0 β i (u)du, 0 T (13) β i () = a i φ i () σ i, 0 T, i = 0, 1,, n. (14), β i () = β i., W i ()., φ i () = φ i, P., φ i /b i = h i, dh i () = (φ i b i h i ())d + σ i d W i (), 0 T. (15) dh i () = b i ( h i h i ())d + σ i d W i (), 0 T. (16) P, h i h i ()., β i. β i = b i σ i (h i h i ), 0 T, i = 0, 1,, n. (17) i = 0, X 0 (, T ) Vasicek [26]. 4.2,,. h i (0 T ),. h i () = h i (0) exp{ b i } + h i (1 exp{ b i }) + σ i exp{ b i ( s)}dw i (s) (18), h i (). 0 E [h i ()] = h i (0) exp{ b i } + h i (1 exp{ b i }), (19) V [h i ()] = σ2 i 2b i (1 exp{ 2b i }) (20) E [h i ()] h i, h h().,. ( T h i (s)ds = h ) T i(u) (1 exp{ b i (T )}) + h i T exp { b i (T s)} ds b i [ T, µ i (, T ) = E µ i (, T ) = h i() b i = h i() b i (1 exp{ b i (T )}) + h i (1 exp{ b i (T )}) + h i 70 + σ T i b i (1 exp{ b i (T s)}) dw i (s) (21) ] h i (s)ds G. ( T T exp { b i (T s)} ds ( T 1 ) (1 exp{ b i (T )}) b i ) (22)
[ ] T, v i (, T ) = V i h i (s)ds G Io s Isomery,. ( v i (, T ) = E σ T 2 i (1 exp{ b i (T s)}) dw i (s)) b i G = σ2 i b 2 i = σ2 i b 2 i T (1 exp{ b i (T s)}) 2 ds ( T + 2 (1 exp{ b i (T )}) 1 ) (1 exp{ 2b i (T )}) b i 2b i (21), OU., [ (0 T ) P[τ i > T F ] = E exp{ ] T h i (s)ds} G. { P[τ i > T F ] = exp µ i (, T ) + v } i(, T ) 2,,. 4.3,.,. OU,,, v ij (, T ) = Cov ij [ T v ij (, T ) = σ i σ j T = ρ ij σ i σ j b i b j h i (s)ds, T ] h j (s)ds G. 1 (1 exp{ b i (T )}) 1 (1 exp{ b j (T )}) ρ ij ds b i b j ( T 1 exp{ b i(t )} 1 exp{ b j(t )} + 1 exp{ (b ) i + b j )(T )} b i b j b i + b j (25) v ii (, T ) = v i (, T ). (7),,. n P[τ 1 > T,... τ n > T F ] = exp µ i (, T ) + 1 n n v ij (, T ) (26) 2 i=1 i=1 j=1 Cov(1 {τi T }, 1 {τj T } F ) Cov(1 {τi>t }, 1 {τj>t } F ), (8) Cov(1 {τi T }, 1 {τj T } F ). { Cov(1 {τi T }, 1 {τj T } F ) = exp (µ i (, T ) + µ j (, T )) + v } i(, T ) + v j (, T ) (exp {v ij (, T )} 1). 2, V ar(1 {τi T } F ) = P[τ i T F ]P[τ i > T F ], (9),. (9),. (23) (24) ρ(1 {τi T }, 1 {τi T } F ) = exp {v ij (, T )} 1 (exp {vii (, T )} 1)(exp {v jj (, T )} 1) (27) 71
4.4 (11) OU h i ().,., h i (). 3. RT, δ i = (1 L i ())X 0 (). L i () = L i. h i. i (i = 1,..., n), ˆr(). Kijima [17], lim T i (, T ) = ˆr i () r 0 (), (6),(24),. ˆr i () r 0 () = (1 δ i )h i () (28) δ i.,,., h i ()., 6, ( ). h i () P., OU. OU, GMM,. h(),, GMM,.,., h() SUR GLS.,,., P h i (). h i ( + ) h i () = b i ( h i h i ()) + σ ϵ i (29), ϵ i (i = 1, 2,..., n) 0, E[ϵ i ϵ j ] = ρ ij (i j). h i0 = h i (), h i1 = h i ( + ),..., h ik = h i ( + k ),..., h ik = h i ( + K ). 1/250.,. h ik+1 h ik = b i ( h i h ik ) + σ i ϵik (i = 1, 2,..., n, k = 0, 1,..., K). (30), h i., h i h i (), h i h ik (k = 0,... K).. Y ik = b i Z ik + u ik (31) Y ik = h ik+1 h ik., Z ik = ( h i h ik ). u ik 0, σi 2.., h() SUR. Y i = Z i b i + u i (i = 1,..., n) E[u ik ] = 0 (i = 1,..., n, k = 0,..., K) E[u 2 ik ] = σ2 i (i = 1,..., n, k = 0,..., K) E[u ik u jk ] = σ i σ j ρ ij (i, j = 1,..., n, k = 0,..., K) E[u ik u jl ] = 0 (if k l) 6,,. (32) 72
Y i = (Y i1,... Y ik ), Z i = (Z i1,... Z ik ), u i = (u i1,... u ik ). Y i Y j (i, j = 1,..., N, i j)., GLS b i, σ i, ρ ij. P h, (9) P 7. β, h i. β,, (24). 5,., Bloomberg,.,,.. δ i = 0.5 (i = 1, 2,..., n)., β i = 0 (i = 0, 1,..., n)., ( ) 8., β = 0, P h() P h().,,,.,,., (McCulloch[23]). 5.1 (11) OU. Bloomberg 1998 9 2003 9.,,,,. 25., 4.., h(), h = 0.0036, b = 25.4669, σ = 0.0038, h = 0.0066, b = 1.3188, σ = 0.0032., ρ = 0.1997. 2 25. 2003 9 1.,, h = 0.0302, b = 2.8461, σ = 0.0138., 2 ρ = 0.4 0.8., BB, h = 0.0858, b = 0.3175, σ = 0.0283, A, h = 0.0058, b = 3.4078, σ = 0.0063.,., BB b,, h()., 5. β = 0, P.,. 7, h. P h h (9),. 8,. 73
0.45 0.4 新日本石油株式会社 麒麟麦酒株式会社 ) ) ( S ( ムアミレプのトーレトッポス 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 001/10/23 001/11/23 001/12/23 002/1/23 002/2/23 002/3/23 002/4/23 002/5/23 002/6/23 002/7/23 002/8/23 002/9/23 002/10/23 002/11/23 002/12/23 003/1/23 003/2/23 003/3/23 003/4/23 003/5/23 003/6/23 003/7/23 003/8/23 003/9/23 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4:. 2001 10 23 2003 9 23. h(), h = 0.0036, b = 25.4669, σ = 0.0038, h = 0.0066, b = 1.3188, σ = 0.0032., ρ = 0.1997. S() = ˆr() r 0 (), S(). 2: Bloomberg 1998 9 2003 9 25. 2003 9 1. (h) (b) (σ).0302 2.8461.0138 A.0058 3.4078.0063 BBB.0155 1.9728.0081 BB.0858 -.3175.0283 74
35 ( % ) 30 全体 A 以上 BBB BB 以下 ) ] T 25 [τ P ( 率 20 確トルォフ15 デ ドイラ10 プンイ 5 0 1 2 3 4 5 期間 ( T ) ( 年 ) 5:. Bloomberg 1998 9 2003 9 25,., h(0) h. 2003 9 1.,,.,. 5.2,,, 1. 25 10 ( ), 10,.,. 3., 4, 5. 0.1 0.6.,, 0.4 0.8. 1.5%, (5),.,,,.,,,.. 5.3 6. 5.. ρ ij ρ(1 {τi T }, 1 {τi T } F ), B. 75
3: Bloomberg 1998 9 2003 9. (h) (b) (σ) (P [τ 1]) 1.0106.4572.0050.0178 2.0060 1.0624.0046.0101 3.0110-1.1132.0052.0272 4.0093.0768.0050.0171 5.0091 1.3629.0062.0137 6.0080 3.3173.0052.0097 7.0085.2988.0050.0126 8.0054 4.1091.0047.0063 9.0081 1.3360.0051.0113 10.0127 -.0011.0052.0208.0089 1.0906.0051.0147 4:. Bloomberg 1998 9 2003 9 25 10. 1 2 3 4 5 6 7 8 9 10 1 1.3559.0962.1736.2817.3779.3657.5539.6835.1528 2.3559 1.1751.1309.6189.6010.6426.4118.3621.4354 3.0962.1751 1.0658.2032.2329.2074.0551.0982.1234 4.1836.1309.0658 1.1029.1039.1719.2796.2553.0999 5.2817.6159.2032.1028 1.5706.5775.2761.2839.4641 6.3779.6010.2329.1039.5706 1.7057.3050.2969.2979 7.3657.6426.2074.1719.5775.7057 1.3971.3661.3947 8.5539.4118.0551.2798.2761.3050.3971 1.7932.2188 9.6835.3621.0982.2553.2839.2969.3661.7932 1.2062 10.1528.4354.1234.0999.4641.2979.3947.2166.2062 1 76
5:. Bloomberg 1998 9 2003 9. 1 2 3 4 5 6 7 8 9 10 1 1.7493.7193.6113.4913.5707.6836.6091.5730.5925 2.7493 1.7381.6131.5546.6410.7696.7248.6385.5575 3.7193.7381 1.6058.4997.5714.7463.6043.5966.5909 4.6113.6131.6058 1.4224.5457.6099.5449.5164.7726 5.4913.5546.4997.4224 1.6232.5314.5811.6285.4098 6.5707.6410.5714.5457.6232 1.6590.7392.7980.5030 7.6836.7696.7463.6099.5314.6590 1.7109.6768.5594 8.6091.7248.6043.5449.5811.7392.7109 1.6999.4844 9.5730.6385.5966.5164.6285.7980.6768.6999 1.4783 10.5925.5575.5909.7726.4098.5030.5594.4844.4783 1 6:. Bloomberg 1998 9 2003 9. 1 2 3 4 5 6 7 8 9 10 1 1.7552.7505.6201.4997.5878.6905.6323.5807.6022 2.7552 1.7800.6232.5602.6511.7766.7402.6427.5686 3.7505.7800 1.6327.5361.6365.7750.6829.6374.6172 4.6201.6232.6327 1.4333.5688.6188.5734.5274.7800 5.4997.5602.5361.4333 1.6329.5411.5937.6337.4215 6.5878.6511.6365.5688.6329 1.6809.7409.8070.5265 7.6905.7766.7750.6188.5411.6809 1.7407.6857.5693 8.6323.7402.6829.5734.5937.7409.7407 1.7124.5121 9.5807.6427.6374.5274.6337.8070.6857.7124 1.4901 10.6022.5686.6172.7800.4215.5265.5693.5121.4901 1 77
6,,,.,..,,.,,.,., OU,,.,.,,.., P GLS,, h., h,.. A (5), (5). RT, (L() = L), δ 1 L,,. X(, T ) = 1 {τ>} Ẽ [ exp { H 0 (, T )} { δ + (1 δ)1 {τ>t } } F ],. X(, T ) = 1 {τ>} Ẽ [ exp { H 0 (, T )} { } ] δ + (1 δ)1 {τ>t } F = 1 {τ>} X 0 (, T ) {δ + (1 δ)ẽ [ ] } 1 {τ>t } F { = 1 {τ>} X 0 (, T ) δ + (1 δ) P } [τ > T F ] { } = 1 {τ>} X 0 (, T ) δ + (1 δ)ẽ [exp { H(, T )} G ] (A1) (7), (7)., T G T, H, G T H, h i ()., {τ i > } (i = 1, 2,..., n),. P[τ i > T G T H ] = exp { H i (, T )} [31]. (0 T ). {τ i > } (i = 1, 2,..., n), P[τ i > T F ] = E [P [τ i > T G T H ] F ] = E [exp { H i (, T )} F ] 78
.,. P[τ 1 > 1,, τ n > n G T H ] = n P[τ i > i G T H ], T,,., Soyanov [24].,. {τ i > } (i = 1, 2,..., n), i=1 P[τ 1 > 1,, τ n > n F ] = E [P[τ 1 > 1,, τ n > n G T H ] F ] [ n ] = E P[τ i > i G T H ] F i=1 [ n ] = E exp { H i (, i )} F i=1 [ { ] n = E exp H i (, i )} F i=1 [ { ] n = E exp H i (, i )} G. (A2), (G T H ) (G H ) = F.,. exp { n i=1 H i(, i )} H. B. i=1 B(b i, b j, σ i, σ j,, T ) σ iσ j b i b j ( T 1 exp{ b i(t )} 1 exp{ b j(t )} + 1 exp{ (b ) i + b j )(T )} b i b j b i + b j, (25),. T 1 v ij (, T ) = σ i σ j (1 exp{ b i (T )}) 1 (1 exp{ b j (T )}) ρ ij ds b i b j ( σ i σ j = ρ ij T 1 exp{ b i(t )} 1 exp{ b j(t )} + 1 exp{ (b ) i + b j )(T )} b i b j b i b j b i + b j ρ ij B(b i, b j, σ i, σ j,, T ) v ii (, T ) = v i (, T )., ρ ij ρ ij., ρ(1 {τi T }, 1 {τi T } F ) = exp {ρ ijb(b i, b j, σ i, σ j,, T )} B(b i, b j, σ i, σ j,, T ) ρ ij (exp {vii (, T )} 1)(exp {v jj (, T )} 1) (B3), ρ ij, ρ ij., B(b i, b j, σ i, σ j,, T ) ρ ij = 1. ρ ij = 1.,,.,, ρ ij 1, ρ ij 1. 79
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