BIS CDO CDO CDO CDO Cifuentes and O Connor[1] Finger[6] Li[8] Duffie and Garleânu[4] CDO Merton[9] CDO 1 CDO CDO CDS CDO three jump model Longstaff an
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1 CDO CDO(Collateralized Debt Obligation) Duffie and Garleânu[4] CDO CDS(Credit Default Swap) Duffie and Garleânu[4] 4 CDO CDS CDO CDS CDO 2007 CDO CDO CDS ( ) CDO CDO 80 taiji.ohka@gmail.com stk25616a@ae.keio.ac.jp 1
2 BIS CDO CDO CDO CDO Cifuentes and O Connor[1] Finger[6] Li[8] Duffie and Garleânu[4] CDO Merton[9] CDO 1 CDO CDO CDS CDO three jump model Longstaff and Rajan[10] CDS CDS CDO Duffie and Garleânu[4] CDO CDS(Credit Default Swap) CDS CDS CDS CDS CDS CDS CDO CDS ISDA(the International Swaps and Derivatives Association) CDS CDS 1 2
3 CDS CDS CDS George[7] CDS CDS CDS Duffie and Singleton[3] RMV(recorvery of market value) RMV CDS RMV CDS CDS 4 CDS CDO CDO ( ) ( ) CDO CDO 2 3 George [7] CDS CDO CDO 2.1 CDS
4 CDS CDS CDS CDS λ L CDS L L CDS Duffie and Singleton[4] RMV RMV T τ ϕ(τ) ϕ(τ) = (1 L)v(τ, T ), λ L CDS CDS George[7] 2 CDS ( ) ( ) CDS r x t(x > t) R(t, x) R(t, x) = x t rdu, t CDS t = 0 t CDS S(t) CDS T (T > 0) 1 1 t = 0 T CDS S(0) CDS S(0) 3 τ(τ < T ) τ R A R A = E Q t [ = S(t)E Q t S(t) ] m exp( R(t, t i ))I {τ>ti } i=1 [ m ] exp( R(t, t i ))I {τ>ti }, (1) 2 CDS 3 n S(t) CDS i=1 4
5 1: CDS (1) E Q t t Q τ R(t, t i ) t i t I {τ>tj } { 1 (τ > t i ) I {τ>ti } = 0 (τ t i ) τ > 0 Y (τ) ϕ(τ) 1 Y (τ) = 1 ϕ(τ), Duffie, Pan, and Singleton[2] 1 T t v(t, T ) T v(t, T ) = exp( R(t, T )) Pr[τ > T t] + (1 L) t exp( R(t, u))f (t, u)du, (2) (2) T 1 f (t, x) t x Pr Q [τ < x t] f (t, x) = d dx Pr [τ > x t] Q t R B t T R B = E Q t [ ] exp( R(t, τ))y (τ)i{τ tj,t j + t}. (3) 5
6 CDS (1) (3) (4) S(t) = EQ t = T t R A = R B, (4) [ exp( R(t, τ))y (τ)i{tj + t>τ>t j }] Q [ ], m exp( R(t, t j ))I {τ>tj } j=1 exp( R(t, x)) {1 (1 L)v(x, T )} f (t, x)dx m, (5) exp( R(t, t j ) Pr Q [τ > t j t] j=1 2.2 Duffie and Garleânu[4] CDS t t t t + t λ λ Pr[τ < t + t t] = λ t. (6) (6) t λ t Pr[τ < t + t t] t t t + t t λ(t) t < τ t + x ( t+x )] Pr[τ < t + x t] = E t [exp λ(u)du, x > 0, t E t t t s λ(t) dλ(t) = κ(θ λ(t))dt + σ λ(t)dw (t) + J(t), (7) W (t) J(t) J(t) µ l κ, θ, σ, µ, l 6
7 κ, θ, σ, µ, l κ θ σ µ l Duffie and Kan[5] t x E Q t (8) [ ( t+x )] exp λ u du = e α(x)+β(x)λ(t), (8) t f (t, x) = d dx Pr[τ > x t] = exp {α(x t) + β(x t)λ(t)} [α (x t) + β (x t)λ(t), ] (9) α(x) β(x) α (x) β (x) α(x) = κθ( c 1 d 1 ) β(x) = b 1 c 1 d 1 1 eb 1 x c 1 +d 1, e b 1 x α (x) = κθβ(x) + l ln c 1+d 1 e b 1x c 1 +d 1 µβ(x) 1 µβ(x), β (x) = κβ(x) σ2 β 2 (x) 1, b 1 = κ 2 + 2σ 2, c 1 = κ+ κ 2 +2σ 2 2, d 1 = κ κ 2 +2σ 2 2, a 2 = d1 c 1, b 2 = b 1, c 2 = 1 µ c 1, d 2 = d 1+µ c 1, + κθ c 1 x + l(a 2c 2 d 2 ) b 2 c 2 d 2 ln c 2+d 2 e b2x c 2 +d 2 + ( l c 2 ) l x, Duffie and Garleânu[4] 1 X κ, θ X, σ, µ, l X Y κ, θ Y, σ, µ, l Y X Y X + Y κ, θ, σ, µ, l θ = θ X + θ Y l = l X + l Y X C(i,j) X C(i) X C X G X C(i,j) (κ, θ XC(i,j), σ, µ, l XC(i,j) ) X C (i) (κ, θ XC(i), σ, µ, l XC (i)) X C (κ, θ XC, σ, µ, l XC ) X G (κ, θ XG, σ, µ, l XG ) X C(i,j) X C(i) X C X G (i = 1, 2,, n, j = 1, 2,, m, C = 1, 2,, l ) λ C(i,j) = X C(i,j) + X C(i) + X C + X G (κ, θ C(i,j), σ, µ, l C(i,j) ) 7
8 X C(i,j) X C(i) X C X G X C(i,j) X C(i) X C X G X C(i,j) C i j 1 X C(i) C i 2 X C C 3 X G 4 ρ C(i,j) = l X G, l C(i,j) (10) ρ C(i,j) = l X C, l C(i,j) (11) ρ C(i,j) = l X C(i) l C(i,j), (12) ρ C(i,j) = l X C(i,j) l C(i,j), (13) E Q t (10) (13) t t + s(s > 0) [ ( t+x )] exp λ C(i,j) (u)du = exp[α(x) + β XC(i,j) (x)x C(i,j) (t) + β XC(i) (x)x C(i) (t) + β XC (x)x C (t) + β XG (x)x G (t)], t α(x) = α XC(i,j) (x) + α XC(i) (x) + α XC (x) + α XG (x). f (t, x) = exp { α(x t) + β C(i,j) (x t)c i,j (t) + β C(i) (x t)x C(i) (t) + β C (x t)x C (t) + β G (x t)x G (t) } [ ] α (x t) + β C(i,j) (x t)x i,j(t) + β C(i) (x t)x C(i)(t) + β C(x t)x C (t) + β G(x t)x G (t), α (x) = α C(i,j) (x) + α C(i) (x) + α C(x) + α G(x), 8
9 2.4 [13] CDO t(t = 0, 1,, N) F S (t) F M (t) F E (t) C S C M t L(t) t Loss S (t) Loss M (t) Loss E (t) (i, j) CDO CDS S i,j (i = 1, 2,, m.j = 1, 2,, n) r (i, j) τ i,j (τ i,j > 0) t + 1 CDO spr(t + 1) m n spr(t + 1) = S i,j I {τi,j >t} + (e r 1)(F S (t) + F M (t) + F E (t)) (F S (t)c S + F M (t)c M ), i=1 j=1 Loss E (t+1) Loss E (t + 1) = min {F E (t) + spr(t + 1), L(t + 1)}, t + 1 F E (t + 1) = F E (t) + spr(t + 1) Loss E (t + 1), Loss M (t + 1) = min {F M (t), max {0, L(t + 1) F E (t) spr(t + 1)}}, t+1 F M (t + 1) = F M (t) Loss M (t + 1), 9
10 Loss S (t + 1) = min {F S (t), max {0, L(t + 1) F E (t) F M (t) spr(t + 1)}}, t + 1 F S (t + 1) = F S (t) Loss S (t + 1), 2.5 CDO CDO CDO (7) i, j λ i,j 4 λ C(i,j) (t) = X C(i,j) + X C (i) + X C + X G, (14) X C(i,j) (t) = X C(i,j) (t t) + κ(θ XC(i,j) X C(i,j) (t t)) t + σ X C(i,j) (t t)ε XC(i,j) t + JXC(i,j) (t), X C(i) (t) = X C(i) (t t) + κ(θ XC(i) X C(i) (t t)) t + σ X C(i) (t t)ε XC(i) t + JXC(i) (t), X C (t) = X C (t t) + κ(θ C X C (t t)) t + σ X C (t t)ε C t + JC (t). X G (t) = X G (t t) + κ(θ G X G (t t)) t + σ X G (t t)ε G t + JG (t). (15) ε XC(i,j) ε XC(i) ε C ε G N(0, 1) J XC(i,j) J XC(i) J C J G J k (t) = { m k l k t 0 1 l k t. m k µ (k = X C(i,j), X C(i), C, G) (15) (14) C(i, j) t t + t (15) CDO X C(i,j), X C(i) 2 10
11 3 CDO CDS CDO 1 4 Quick CDS CDS CDS 3 CDS Quick r 1 L Rating and Investment Information, Inc [12] L = 0.95 CDS T T = R I A ( 5405) R I AA Rating and Investment Information,Inc [12] R I CDS CDS CDS 2009 CDS 3 CDS B
12 2: CDS ( :bbs) t p [0, p] Nelsen[11] Li[8] CDO MATLAB CDO ,2 70 0, L 95 r 1 CDO 72 CDS CDO 5 CDO 3.2 (15) CDO 12
13 (10) ρ C(i,j) ρ C(i,j) (11) 3 CDS ( ) CDS ( ) (11) (12) (13) 3 ρ C(i,j) ρ C(i,j) 3 ρ C(i,j) ρ C(i,j) JAL ρ C(i,j) ρ C(i,j) 4 ρ C (i, j) 4 ρ C (i, j) 3 ρ C(i,j) ρ C (i, j) 5 5 ρ C(i,j) ρ C(i,j) 4 ANA ρ C(i,j) ρ C(i,j) JAL ρ 13
14 3: 14
15 4: 15
16 5: 16
17 3.3 CDO CDO CDO ( ) ( ) ( ) CDS 2 1: (2007/12/ /12/9) CCC B (2006/9/ /12/9) CCC BB 2, (2007/12/ /12/9) CCC AAA AAA AAA 1 (2007/12/ /12/9) (2006/9/ /12/9) 2 ( ), R I CDO
18 6: BBB
19 7: ( ) 8: ( ) 19
20 10 9 9: 2 ( ) CDO
21 10: 2 ( ) 11: ( ) 21
22 : ( ) 2: ( ) (2007/12/ /12/9) AA AAA
23 AAA 4 CDS CDO CDO CDO CDO CDO CDO [1] Cifuentes,A. and G. O Connor. The Bionomial Expanision Method Applied to CBO/CLO Analysis. Moody sinvestors Service, [2] Duffie, D., J. Pan, and K. Singleton. Transform Analysis and Asset Pricing for Affine Jump Diffusion. Econometrica, Vol.12, pp , [3] Duffie, D. and K. Singleton. Modeling term Structures of Defaultable Bonds. Review of Financial Studies, [4] Duffie, D., N. Garleânu.Risk and Valuation of Collateralized Debt Obligations. Financial Analysts Journal, January-February, pp.41-59, [5] Duffie, D., R. Kan. A Yield Factor Model of Interest Rates.Mathematical Finance, Vol.6, pp ,
24 [6] Finger, C. C.A Comparison of Stochastic Default Rate Models. Working Paper, The RiskMetrics Group, [7] George Chacko, Anders Sjoman, Hideto Motohashi, Vincent Dessain. Credit Derivative: A Primer on Credit Risk, Modeling, and Instruments,Wharton School Publishing, 2006.(,,,,,,2008.) [8] Li, D. X. On Default Correlation:A Copula Approach. The Journal of Fixed Income, Vol.9, pp.43-54, [9] R, C, Merton. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. Journal of Finance, Vol.29, pp , [10] Longstaff,Francis A. Arivind Rajan. An Empirical Analysis of the Pricing of Collateralized Debt Obligations. Journal of Finance,2007. [11] Nelsen, R. B. An Introduction to Copulas, Springer, New York, [12] Rating and Investment Information, Inc. R I Tranche Pad Version 1.0 Technical Document.R I, [13],. -ABS CDO., A 3: ( : : http : // i.co.jp/jpn/news t opics/detail/200906/j09 a 066a.pdf) AAA AA A BBB BB B : 72 (R I : ) AAA AA A BBB B
25 5: R I 72 ( ) A A BBB A A AA A AA A AA A A AA A A A A A AA A NEC A A A A AA A AA A AA NTT AA BBB KDDI A A NTT A AA A AA A JR AA JT A BBB AA BBB AA BBB AA AA A A A AAA A BBB A A A BBB AA A BBB AA ANA BBB JFE AA JAL CCC AA A A A AA A A A 25
26 B 6: (r = 0.01 L = 0.95 T = 1800) : σ µ κ l C θ C RSME 4.560E E E E E E-04 7: (r = 0.01 L = 0.95 T = 1800) : θ C(i) l C(i) 4.700E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05 26
27 8: (r = 0.01 L = 0.95 T = 1800) : l C(i,j) θ C(i,j) 1.755E E E E E E E E E E-06 JT 2.593E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-06 NEC 2.716E E E E E E E E E E E E E E E E E E E E E E E E E E-06 JFE 4.185E E E E E E E E E E E E-06 ANA 1.514E E E E-06 JAL 1.449E E E E E E E E E E E E E E-06 NTT 2.506E E E E-06 KDDI 2.591E E E E-06 NTT 2.505E E E E-06 27
28 9: (r = 0.01 L = 0.95 T = 1800) σ µ κ l C θ C RSME E E E E E-5 10: (r = 0.01 L = 0.95 T = 1800) θ C(i) l C(i) 1.450E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05 28
29 11: (r = 0.01 L = 0.95 T = 1800) l C(i,j) θ C(i,j) 9.290E E E E E E E E E E-06 JT 8.727E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-06 NEC 8.872E E E E E E E E E E E E E E E E E E E E E E E E E E-06 JFE 8.689E E E E E E E E E E E E-06 ANA 7.420E E E E-06 JAL 1.607E E E E E E E E E E E E E E-06 NTT 8.520E E E E-06 KDDI 8.660E E E E-06 NTT 8.510E E E E-06 29
30 12: (r = 0.01 L = 0.95 T = 1800) σ µ κ l C θ C RSME 9.999E E E E E E-04 13: (r = 0.01 L = 0.95 T = 1800) θ C(i) l C(i) 4.367E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-06 30
31 14: (r = 0.01 L = 0.95 T = 1800) l C(i,j) θ C(i,j) 3.328E E E E E E E E E E-05 JT 1.702E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05 NEC 3.051E E E E E E E E E E E E E E E E E E E E E E E E E E-05 JFE 1.483E E E E E E E E E E E E-05 ANA 6.980E E E E-05 JAL 7.642E E E E E E E E E E E E E E-05 NTT 8.070E E E E-05 KDDI 2.174E E E E-05 NTT 8.072E E E E-05 31
32 15: 2 (r = 0.01 L = 0.95 T = 1800) sigma mu kappa l2 theta2 RSME E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-04 32
33 16: 2 (r = 0.01 L = 0.95 T = 1800) l C(i,j) θ C(i,j) 2.331E E E E E E E E E E-04 JT 1.202E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-06 NEC 1.262E E E E E E E E E E E E E E E E E E E E E E E E E E-05 JFE 1.066E E E E E E E E E E E E-05 ANA 6.131E E E E-05 JAL 6.986E E E E E E E E E E E E E E-05 NTT 1.057E E E E-05 KDDI 1.170E E E E-05 NTT 1.056E E E E-05 33
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