41, 1, 2011 9 1 21 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. 113-0033 7-3-1 (E-mail: hiroumi.misaki@gmail.com). 1) (2008)
2 41 1 2011 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 Takahashi and Sao (2001) Zhou e al. (2008) CIR Takahashi and Sao (2001) 2. 2.1 1 Björk (2004) (Ω, F, P ) d X dx = µ(, X )d + σ(, X )db (2.1)
4 41 1 2011 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 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 41 1 2011 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 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 ) 4 3. 3.1 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 41 1 2011 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 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 41 1 2011 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) 4. 4.1 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 1 + 1 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 1 + 1 ρ 2 v 3 ] X = F (X, v )
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 = 0.0005 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 41 1 2011 0.020 0.015 L6m S2y S5y L12m S3y S7y 0.010 0.005 1999 2000 2001 2002 2003 2004 2005 2006 2007 1.10 Bond 1.05 1.00 0.95 1999 2000 2001 2002 2003 2004 2005 2006 2007 6 12 LIBOR L6m, L12m 2, 3, 5, 7 S2y, S3y, S5y, S7y Bond 1 1 LIBOR LIBOR 6 12 2 3 5 7 7 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 ) 18 4.2 1996 2006 12 6 2.9% 1 Y 7 6 12 LIBOR 2 3 5 7 1998 11 2006 11 418 12) 1 11) 12) Daasream Bloomberg
13 4.3 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) 1. 2. 3. 13) Takahashi and Sao (2001) (2002) 14)
14 41 1 2011 0.01 X1 0.00 0.01 1999 2000 2001 2002 2003 2004 2005 2006 2007 0.050 X2 0.025 0.000 0.025 1999 2000 2001 2002 2003 2004 2005 2006 2007 0.03 X3 0.02 0.01 1999 2000 2001 2002 2003 2004 2005 2006 2007 ±2σ 2 4. NM 5. NM NM (M, J) = (5000, 20) (50000, 100) (M, J) = (5 10 5, 300) PRIMERGY RX200S5 15) 4.4 2 LIBOR 3 3 2 1 15) Ox 5.10 OxMPI (Ox Doornik (2007) )
15 0.005 X1 0.000 0.005 0.010 1999 2000 2001 2002 2003 2004 2005 2006 2007 0.006 L6m 0.004 0.002 1999 2000 2001 2002 2003 2004 2005 2006 2007 0.025 X2 0.000 0.025 1999 2000 2001 2002 2003 2004 2005 2006 2007 0.020 S5y S7y 0.015 0.010 0.005 1999 2000 2001 2002 2003 2004 2005 2006 2007 3 1 6 LIBOR 2 5 7 2001 10 11 16) 2002 3 400 17) 1 1 2 Î(ˆθ) 1/2 3 418 ɛ 0.0056 16) 2001 11 11 17) 2001 11 16
16 41 1 2011 L6m 0.0075 L12m 0.0100 S2y 0.0050 0.0050 0.0075 0.0050 0.010 S3y 0.020 0.015 S5y S7y 0.02 0.005 0.010 0.005 0.01 1.10 Bond 1.05 1.00 0.95 4 0.0050 L6m 0.0050 L12m 0.0050 S2y 0.0000 0.0000 0.0000 0.0050 S3y 0.0050 S5y 0.0050 S7y 0.0000 0.0000 0.0000 0.025 Bond 0.000 0.025 5
17 1 a b c θ 2 θ 3 0.29 0.18 0.077 0.039 0.011 0.29 0.25 0.058 0.024 0.020 (0.014) (0.012) (0.0028) (0.0012) (0.00098) σ 1 σ 12 σ 2 σ 3 ρ ɛ 0.0049 0.018 0.025 0.051 0.0052 0.0056 0.0048 0.032 0.014 0.036 0.54 0.0047 (0.00023) (0.0017) (0.00068) (0.0018) (0.026) (0.00023) σ u1 σ u2 σ u3 σ u4 σ u5 σ u6 σ u7 0.00025 0.000023 0.00022 0.00011 0.00073 0.0014 0.00049 0.000077 0.000024 0.00014 0.00013 0.00057 0.0012 0.00057 (0.38) (0.12) (0.68) (0.64) (2.8) (5.9) (2.8) 1 2 Î(ˆθ) 1/2 3 ) 3 3 10 5 1 2 1 ρ 2 θ 2 0.039 X 2 4 5 5.
18 41 1 2011 18) 2 22 9712 22 0010 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) 3.1 4.3 2.(c)
19 A1 (A) φ, T ˆφ (B) (C) φ 0 0.3 0.5 0.7 T 100 200 400 1000 100 200 400 1000 100 200 400 1000 100 200 400 1000 (A) 0.00 0.00 0.00 0.00 0.27 0.28 0.29 0.30 0.47 0.48 0.49 0.50 0.67 0.69 0.69 0.70 0.00 0.00 0.01 0.00 0.28 0.21 0.16 0.10 0.32 0.23 0.16 0.10 0.29 0.20 0.14 0.08 1.80 1.89 1.94 1.97 1.65 1.68 1.67 1.67 1.37 1.33 1.30 1.28 0.98 0.93 0.90 0.88 (B) 2.02 2.01 2.00 2.00 1.78 1.73 1.70 1.69 1.40 1.34 1.30 1.28 0.97 0.92 0.89 0.88 (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 = 5000 1.61 1.60 1.57 1.58 1.29 1.34 1.40 1.44 1.29 1.27 1.24 1.23 0.96 0.91 0.88 0.87 (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 = 20000 1.90 1.87 1.86 1.86 1.51 1.55 1.60 1.61 1.35 1.33 1.28 1.27 0.97 0.92 0.89 0.88 (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 41 1 2011 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 = 10 5 3 Î(φ) k ˆφ (k) Î( ˆφ (k) ) {Î( ˆφ(k) ) 1 2 k=1 A1(B) T T = 100 19) K = 2000 M = 5000, 20000 2 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. 267 281, 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), 143 172. Cox, J., Ingersoll, J. and Ross, S. (1985). A Theory of he Term Srucure of Ineres Raes, Economerica, 53, 385 408. 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), 197 226. Duffie, D. and Singleon, K. J. (1999). Modeling Term Srucures of Defaulable Bonds, The Review of Financial Sudies, 12(4), 687 720. Higuchi, T. and Kiagawa, G. (2000). Knowledge Discovery and Self-Organizing Sae Space Model, IEICE Transacions on Informaion and Sysems, E83-D(1), 36 43. Hull, J. and Whie, A. (1994). Numerical Procedures for Implemening Term Srucure Models II: Two-Facor Models, J. Deriv., 2, 37 48. Hull, J. and Whie, A. (1997). Taking Raes o he Limis, Risk, December, 168 169. Kiagawa, G. (1996). Mone Carlo Filer and Smooher for Non-Gaussian Nonlinear Sae Space Models, J. Compu. Graph. Sa., 5(1), 1 25. Kiagawa, G. (1998). A Self-Organizing Sae-Space Model, J. Am. Sa. Assoc., 93, 1203 1215. (2001) } K 19) ±10% φ = 0 ±0.05
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