min(v T, D) 2) 3) inensiy model (defaul inensiy) (implied defaul inensiy) (CDS) 4) Duffie and Singleon (1999) Duffee (1999) CIR (Cox e al.

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1 41, 1, Credi Risk Esimaion wih a Paricle Filer Hiroumi Misaki Takahashi and Sao (2001) This paper is developing a new esimaion mehod of he defaul inensiy implied in he price of defaulable corporae bonds. We formulae he reduced form credi risk model as a nonlinear sae space model and use he paricle filering mehod o esimae he defaul inensiy. We exend he erm-srucure model proposed by Takahashi and Sao (2001) and incorporae he defaul inensiy. By using he Japanese corporae bond marke, we shall show ha our esimaion mehod gives a reasonable resul. :. 1. 1) Meron (1974) opion based model V D T T DC ( 1) (2008)

2 min(v T, D) 2) 3) inensiy model (defaul inensiy) (implied defaul inensiy) (CDS) 4) Duffie and Singleon (1999) Duffee (1999) CIR (Cox e al. (1985)) 3 3 r λ Chen and Sco (2003) CIR r λ 2) T D V T 3) 4) CDS ( (2001) )

3 3 Takahashi and Sao (2001) Zhou e al. (2008) CIR Takahashi and Sao (2001) Björk (2004) (Ω, F, P ) d X dx = µ(, X )d + σ(, X )db (2.1)

4 B n µ, σ d 1 d n r(x ) X T P (, X ; T ) 1 { ] T P (, X ; T ) = E [exp Q r(x s )ds} F (2.2) E Q Q F Q dx = {µ(, X ) ψ(, X )}d + σ(, X )d B d 1 ψ(, X ) B Q 2.2 Lando (2004, Chaper 5) (hazard rae) τ F () = P [τ ] F () h() = d d log F () ( P [τ ] = exp h lim 0 0 ) h(s)ds 1 P [τ + τ > ] = h() h() (, + ] λ P [τ + F ] I {τ>} λ (2.3) I F λ F - λ (defaul inensiy).

5 5 λ X N := I {τ } N Cox 5) Cox λ Cox (Ω, F, Q) X R d λ : R d R G := σ{x s : 0 s } (G ) 0 Exp(1) E 1 H := σ{n s : 0 s } F := G H F τ τ := inf { : 0 λ(x s )ds E 1 } N τ λ(x ) (, + ] λ(x ) Cox T 1 V (, X ; T ) Q τ λ(x ) { P (, X ; T ) = E [exp Q } ] T F r(x u )du r(x s ) 6) 0 ( )] V (0, X 0 ; T ) = E Q [exp T 0 (r + λ)(x s )ds 7) Cox 5) Cox Cox 6) τ n H E hexp Q R o i h n T r(x u)du F = E Q exp R o i T r(x u)du G (McNiel e al. (2005) ) 7) Lando (2004) r(x s )+λ(x s ) (r+λ)(x s ) R r(x s )+(1 R)λ(X s ) (r + (1 R))λ(X s)

6 τ ( Q(τ > T G T ) = exp T 0 ) λ(x s )ds ({τ > }) ( V (, X ; T ) = E [exp Q T (r + λ)(x s )ds) G ] (2.4) E Q 2.3 (recovery rae) R (i) (Recovery of face value) (ii) (Recovery of marke value; RMV) (iii) (Recovery of reasury) (Schönbucher (2003)) (i) R (ii) R (iii) R (ii) RMV RMV R [0, 1) Duffie and Singleon (1999) T E Q [ { exp T }] (r + (1 R)λ) (X s )ds (2.5) R λ (2.5) (1 R)λ λ (2.4) λ

7 7 2.4 P (, X ; T ) V (, X ; T ) X x = f(x 1, v ) (2.6) y = h(x, w ) (2.7) x y v q(v) w = 1,..., T y x {y 1,..., y } x x X (2.1) (2.6) y P (, X ; T ) V (, X ; T ) P (, X ; T ) V (, X ; T ) X (2.2), (2.4) X (2.7) λ(x ) X X λ(x ) Kiagawa (1996) p (1),..., p (M) p(x Y 1: 1 ) f (1),..., f (M) p(x Y 1: ) s (1) T,..., s(m) T p(x Y 1:T )

8 Y 1: := {y 1,..., y } M Kiagawa (1996) 1. p 0 (x) j = 1,..., M f (j) 0 2. = 1,..., T (a) j = 1,..., M v (j) (b) j = 1,..., M p (j) (c) j = 1,..., M α (j) (d) j = 1,..., M f (j) f (j) = q(v) = f(f (j) 1, v(j) ) = p(y p (j) ) p (1) w.p. α (1) / M. p (M). i=1 α(i) w.p. α (M) / M i=1 α(i) θ p(y Y 1: 1 ) = p(y x )p(x Y 1: 1 )dx 1 M M i=1 p(y p (i) ) = 1 M M i=1 α (i) l(θ) = T log p(y Y 1: 1 ) =1 ( T M log =1 i=1 α (i) ) T log M. (3.1) (3.1) ˆθ AIC (Akaike (1973)) AIC AIC = 2l(ˆθ) + 2( )

9 9 3.2 (3.1) Kiagawa (1998) Kiagawa (1998) θ z = [x, θ] (2.6), (2.7) z = F (z 1, v ) = [f(x 1, v ; θ), θ] y = H(z, w ) = h(x, w ; θ) 8) v 2 θ = θ 1 + v 2 z = [x, θ ] (Kiagawa (1998), Higuchi and Kiagawa (2000) ) v 2 9) v 2 (M = 10 6 ) Lin e al. (2004) Nelder-Mead Nelder and Mead (1965) NM NM (3.1) 10) NM 8) (10 8 ) Nakamura e al. (2009) 9) Liu and Wes (2001) 10) Yano (2007) NM

10 ˆθ I(θ) T (ˆθ θ) N(0, I(θ) 1 ) I(θ) Î(ˆθ) ij = 1 T T log p(y Y 1 ; θ) log p(y Y 1 ; θ) θ i θ= ˆθ θ j =1 θ= ˆθ ˆθ Î(ˆθ) Hull and Whie (1994, 1997) Takahashi and Sao (2001) 2 3 CIR X = (X 1, X 2, X 3 ) dx 1 = a(x 2 X 1 )d + σ 1 db 1 dx 2 = b(θ 2 X 2 )d + σ 12 db 1 + σ 2 db 2 dx 3 = c(θ 3 X 3 )d + σ 3 X3 [ ρdb ρ 2 db 3 ] 1 2 X 1, = X 1, + a(x 2, X 1, ) + σ 1 v 1 X 2, = X 2, + b(θ 2 X 2, ) + (σ 12 v 1 + σ 2 v 2 ) X 3, = X 3, + c(θ 3 X 3, ) + σ 3 X3, [ρv ρ 2 v 3 ] X = F (X, v )

11 11 X 1 X 1 ɛ r(x ) = ɛ exp { X 1 ɛ } X1 < ɛ ɛ 3 λ(x ) = X 3 Takahashi and Sao (2001) ɛ = ɛ r λ T X [ { }] P (, X ; T ) = E Q exp T r(x s )ds (4.1) [ { V (, X ; T ) = E Q exp T (r + λ)(x s )ds }] (4.2) (4.1) (4.2) LIBOR τ n LIBOR ( ) 1 1 L (X, τ n ) = P (, X ; + τ n ) 1 τ n δ S (X, τ n ) = 1 P (, X ; + τ n ) δ τ n /δ i=1 P (, X ; + iδ) (Björk (2004) ) T j C V (X, T, C, { j }) τ n V (X, T, C, { j }) = C j V (, X ; j ) + V (, X ; T ).

12 L6m S2y S5y L12m S3y S7y Bond LIBOR L6m, L12m 2, 3, 5, 7 S2y, S3y, S5y, S7y Bond 1 1 LIBOR LIBOR Y L (X, τ n ) + u n,, n = 1, 2, Y n, = S (X, τ n ) + u n,, n = 3,..., 6, V (X, T, C, { j }) + u n,, n = 7 (4.3) (τ 1,..., τ 6 ) = (0.5, 1, 2, 3, 5, 7) u n, Σ u = diag(σu1, 2..., σu7) 2 11) (4.3) Y = H(X ) + u θ = (a, b, c, θ 2, θ 3, σ 1, σ 12, σ 2, σ 3, ρ, ɛ, Σ u ) % 1 Y LIBOR ) 1 11) 12) Daasream Bloomberg

13 X = F (X, v ) Y = H(X ) + u [0, T ] 1. 3 p 0 (x) j = 1,..., M f (j) 0 2. =, 2,..., T, T (a) j = 1,..., M v (j) (b) j = 1,..., M p (j) = F (f (j), v(j) 3 N 3 (0, I) ) (c) j = 1,..., M 7 N 7 (0, Σ u ) Y H(p (j) ) α (j) H(p (j) ) P (, p (j) ; ) V (, p (j) ; ) p (j) J 13) (d) j = 1,..., M f (j) α (j) 1 14) ) Takahashi and Sao (2001) (2002) 14)

14 X X X ±2σ 2 4. NM 5. NM NM (M, J) = (5000, 20) (50000, 100) (M, J) = (5 10 5, 300) PRIMERGY RX200S5 15) LIBOR ) Ox 5.10 OxMPI (Ox Doornik (2007) )

15 X L6m X S5y S7y LIBOR ) ) Î(ˆθ) 1/ ɛ ) )

16 L6m L12m S2y S3y S5y S7y Bond L6m L12m S2y S3y S5y S7y Bond

17 17 1 a b c θ 2 θ (0.014) (0.012) (0.0028) (0.0012) ( ) σ 1 σ 12 σ 2 σ 3 ρ ɛ ( ) (0.0017) ( ) (0.0018) (0.026) ( ) σ u1 σ u2 σ u3 σ u4 σ u5 σ u6 σ u (0.38) (0.12) (0.68) (0.64) (2.8) (5.9) (2.8) 1 2 Î(ˆθ) 1/2 3 ) ρ 2 θ X

18 ) A. A.1 AR(1) + φ ( φ < 1) X = φx 1 + ζ, ζ i.i.d. N(0, 1), Y = X + η, η i.i.d. N(0, 1), = 1,..., T. T ( ˆφ φ) Y := {Y 1, Y 2,..., Y T } k Y (k), k = 1,..., K Y (k) ˆφ (k) 18) (c)

19 19 A1 (A) φ, T ˆφ (B) (C) φ T (A) (B) (0.38) (0.26) (0.18) (0.11) (0.40) (0.28) (0.20) (0.13) (0.36) (0.24) (0.17) (0.10) (0.26) (0.16) (0.11) (0.07) (C) M = (0.30) (0.22) (0.15) (0.11) (0.42) (0.32) (0.18) (0.10) (0.29) (0.20) (0.14) (0.09) (0.24) (0.16) (0.11) (0.07) M = (0.37) (0.24) (0.17) (0.11) (0.40) (0.31) (0.18) (0.11) (0.32) (0.23) (0.16) (0.10) (0.25) (0.16) (0.11) (0.07) 1) (A)-(C) φ 4 T 2) (B),(C) nî( ˆφ(k) ) 1 2 o K k=1

20 Y (k) ˆφ (k) K { ˆφ (k) } K k=1 ˆφ φ = 0, 0.3, 0.5, 0.7 A1(A) 1 ˆφ 2 T ( ˆφ φ) 3 T ( ˆφ φ) K = Î(φ) k ˆφ (k) Î( ˆφ (k) ) {Î( ˆφ(k) ) 1 2 k=1 A1(B) T T = ) K = 2000 M = 5000, A1(C) Akaike, H. (1973). Informaion Theory and an Exension of he Maximum Likelihood Principle, 2nd Inernaional Symposium on Informaion Theory (eds. B. N. Perov and F. Csáki), pp , Akadémiai Kiadó, Budapes. Björk, T. (2004). Arbirage Theory in Coninuous Time, 2nd ed., Oxford Universiy Press. Chen, R. and Sco, L. (2003). Muli-Facor Cox-Ingersoll-Ross Models of he Term Srucure, J. Real Esae Financ. Econ., 27(2), Cox, J., Ingersoll, J. and Ross, S. (1985). A Theory of he Term Srucure of Ineres Raes, Economerica, 53, Doornik, J. A. (2007). Objec-Oriened Marix Programming Using Ox, 3rd ed., London, Timberlake Consulans Press and Oxford. Duffee, G. R. (1999). Esimaing he Price of Defaul Risk, The Review of Financial Sudies, 12(11), Duffie, D. and Singleon, K. J. (1999). Modeling Term Srucures of Defaulable Bonds, The Review of Financial Sudies, 12(4), Higuchi, T. and Kiagawa, G. (2000). Knowledge Discovery and Self-Organizing Sae Space Model, IEICE Transacions on Informaion and Sysems, E83-D(1), Hull, J. and Whie, A. (1994). Numerical Procedures for Implemening Term Srucure Models II: Two-Facor Models, J. Deriv., 2, Hull, J. and Whie, A. (1997). Taking Raes o he Limis, Risk, December, Kiagawa, G. (1996). Mone Carlo Filer and Smooher for Non-Gaussian Nonlinear Sae Space Models, J. Compu. Graph. Sa., 5(1), Kiagawa, G. (1998). A Self-Organizing Sae-Space Model, J. Am. Sa. Assoc., 93, (2001) } K 19) ±10% φ = 0 ±0.05

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