デフォルト相関係数のインプライド推計（）

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1 ,,,.,.,.,,.,,.,,.,,. :,, OU, SUR,.,.,,.,.., 62

2 1, BIS,.,.,,,,.,,.,.,, [33],,,.,,.,,,,.,,.,,,.. 2,. 3,. 4,,. 5. 2,,.,. 2.1,.,,.,. Giesecke [10] dependen defaul.,,,, GDP,.,,.,...,,,.,.,.,,.,,. Giesecke [10], 1. 63

3 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,., Meron [22].. 64

4 ,.,.,,.,.,,, (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

5 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

6 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

7 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

8 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,,

9 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)

10 [ ] 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

11 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

12 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 ,,,,. 25., 4.., h(), h = , b = , σ = , h = , b = , σ = , ρ = ,, h = , b = , σ = , 2 ρ = , BB, h = , b = , σ = , A, h = , b = , σ = ,., BB b,, h()., 5. β = 0, P.,. 7, h. P h h (9),. 8,. 73

13 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 /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/ : h(), h = , b = , σ = , h = , b = , σ = , ρ = S() = ˆr() r 0 (), S(). 2: Bloomberg (h) (b) (σ) A BBB BB

14 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 プンイ期間 ( T ) ( 年 ) 5:. Bloomberg ,., h(0) h ,,.,. 5.2,,, ( ), 10,.,. 3., 4, ,, %, (5),.,,,.,,, ρ ij ρ(1 {τi T }, 1 {τi T } F ), B. 75

15 3: Bloomberg (h) (b) (σ) (P [τ 1]) :. Bloomberg

16 5:. Bloomberg :. Bloomberg

17 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

18 .,. 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

19 [1] Aonuma, K., and Nakagawa, H., Valuaion of Credi Defaul Swap and Parameer Esimaion for Vasicek-pe Hazard Rae Model., Working paper. [2] Basel Commiiee on Banking Supervision. (1999). Credi Risk Modelling: Curren Pracice and Applicaion. [3] Basel Commiiee on Banking Supervision. (2003). The New Basel Capial Accord, Third Consulaive Paper. [4] Collin-Dufresne,P., R.S.Goldsein, and J.Hugonnier. (2003) A general formula for valuing defaulable securiies. Working paper, CMU. [5] Duffie, D. (1998) Firs-o-Defaul Valuaion. Working paper, Sanford Universiy. [6] Duffie, D., J. Pan, and Singleon, K.. (2000) Transform Analysis and Asse Pricing for Affine Jump Diffusions. Economerica, 68, [7] Duffie, D., and Singleon, K. (1999). Modeling erm srucures of defaulable bonds. Review of Financial Sudies, 12, [8] Duffie, D., and Singleon, K. (1999). Simulaing credi correlaion., Workingpaper, GSB, Sanford Universiy. [9] Duffie, D., and Singleon, K. (2003). Credi Risk: Pricing, Managemen, and Measuremen (Princeon Series in Finance), Princeon Universiy Prress. [10] Giesecke, K. (2004). Credi risk modeling and valuaion: an inroducion in Credi Risk: Models and Managemen, 2, Edied by D. Shimko, Riskbooks, London. [11] Giesecke, K., and Weber, S. (2003). Cyclical correlaions, credi conagion, and porfolio losses. o appear in Journal of Banking and Finance. [12] Giesecke, K., and Weber, S. (2003). Correlaed defaul wih imcomplee informaion. o appear in Journal of Banking and Finance. [13] Gordy, M.B. (2003). A risk-facor model foundaion for raing-based bank capial rules. Journal of Financial Inermediaion, 12, [14] Jacobson, T., and Roszbach, K. (2003). Bank lending policy, credi scoring and value-a-risk. Journal of Banking and Finance, 27, [15] Jarrow, R.A., and Turnbull, S.M. (1995). Pricing derivaives on financial securiies subjec o credi risk. Journal of Finance, 50, [16] JP Morgan. (1997). CrediMerics Technical Documen. JP Morgan, New York. [17] Kijima, M. (1999). A gaussian erm srucure model of credi risk spreads and valuaion of yield-spread opions., Workingpaper, Tokyo Meropolian Universiy. [18] Kijima, M. (2000). Valuaion of a credi swap of he baske ype. Review of Derivaives Research, 4, [19] Kijima, M., and Muromachi,Y. (2000). Credi evens and he valuaion of credi derivaives of baske ype. Review of Derivaives Research, 4, [20] Kijima, M., and Muromachi,Y. (2000). Evaluaion of credi risk of a porfolio wih sochasic ineres rae and defaul processes. Journal of Risk, 3(1),5-36. [21] Kusuoka, S., A remark on defaul risk models. Advances in Mahemaical Economics 1, 69-82, 1999 [22] Meron, R. (1974). On he pricing of corporae deb: The risk srucure of ineres raes. Journal of Finance, 29,

20 [23] McCulloch, J.H. (1971) Measuring he erm srucure of ineres raes. Journal of Business, 44, [24] Soyanov, J. (1987). Counerexamples in Probabiliy. Wiley, New York. [25] M. Jeanblanc and Rukowski, M. (2000). Modeling defaul risk: Mahemaical ools. Universie d Evry and Warsaw Universiyof Technology. [26] Vasicek, Oldrich. (1977). An Equilibrium Characerizaion of he Term Srucure, Journal of Financial Economics, 5, [27],. (1999).,. [28]. (2004).,. [29],,. (2001).,. [30]. (2002).., 50(2), [31],, 2001 [32],, 1991 [33],,. (2003)... 81

○松本委員

○松本委員 CIRJE-J-100 2003 11 CIRJE hp://www.e.u-okyo.ac.jp/cirje/research/03research02dp_j.hml Credi Risk Modeling Approaches 2003 11 17 Absrac This aricle originaes from a speech given by he auhor in he seminar

デフォルト相関係数のインプライド推計（ ）

デフォルト相関係数のインプライド推計（）