IMES DISCUSSION PAPER SERIES Discussion Paper No. 99-J- 9 -J-19 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN 100-8630 03
IMES Discussion Paper Series 99-J- 9 -J-19 1999 6 * * [1999] *(E-mail: akira.ieda@boj.or.jp, toshinao.yoshiba@boj.or.jp)
. JLT
Longstaff and Schwartz [1995]LS Jarrow, Lando and Turnbull [1997] Kijima and Komoribayashi [1998] JLT [1999] Moody s 1 [1999] LS 3 JLT 1 [1999] Moody s AA A
LS V K V r s Z 1 dv = rvdt + svdz 1 (1) r a, b h Z Vasicek dr = ( a - br) dt + hdz () r dz1 dz = rdt (3) H (structural model) (reduced form model) Jarrow, Lando and Turnbull [1997] Duffie and Singleton [1998] default probability, default intensity [1999]
s V H V + rshv H V r h + H r + rv H V H + ( a - br) r - rh H = t (4) (4) 3 X = V / K T P t ( X, r, T) =1 w D( r, T ) Q( X, r, T ) 4 P t ( X, r, T ) = D( r, T) - wd( r, T ) Q( X, r, T ) (5) D( r, T ) D( r, T ) = exp( A( T ) - B( T ) r) A(T ) B(T ) æ h A( T ) = ç è b a ö - T b ø æ h a ö æ h ö + ç - exp( 3 ç 3 4 T è b b ø è b ø ( -bt) -1)- ç ( exp( -b ) -1) 1- exp( -b T ) B( T ) = b (5) Q( X, r, T ) 3 4 Longstaff and Schwartz [1995] P t ( X, r, T) Q( X, r, T ) P t Q
Q( X, r, T, n) n 5 Q( X, r, T, n) = n å q i i= 1 q i q = N( 1 ) 1 a i 1 - å - qi = N( ai ) q j N( bij ) i =,3, K, n j= 1 N( ) a i b ij - ln X - M ( it / n, T ) a i = S( it / n) M ( jt / n, T ) - M ( it / n, T ) b ij = S( it / n) - S( jt / n) M ( t, T ) S(t) æa - rsh h M ( t, T) = ç - è b b s - ö t ø æ rsh h ö + ç + exp( - ) - 3 bt bt è b b ø æ r a + ç - è b b h b ö ø + t 3 ( 1- exp( -b )) æ h ö - ç exp( -bt ) bt b 3 è ø æ rsh h S( t) = ç + + s è b b ö t ø æ rsh h ö - ç + 3 b b è ø ( exp( ) 1) ) ( 1- exp( - )) ( 1- exp( -b )) t 5 Longstaff and Schwartz [1995] n=00
æ h + ç è b 3 ö ø ( 1- exp( -bt)) LS 6 YTM LS YTM YTM JGB (YTM) JGB 6 a, b, h s, X, w r r T
7 98 10 9 R&I 8 [ 1] 9 AAA AA A BBB BB B [ ]JGB bps 3,500 3,000,500,000 1,500 1,000 AAA AA A BBB BB B 500 0 0 4 6 8 10 1 LS () () dr = ( a - br) dt + hdz () O/N 97 10 1 7 8 AA A 9 B 1 B
98 10 8 98 10 9 a, b, h 98 10 9 Libor r [ 3] a b h r 0.04748 94.903 0.009149 0.003639 Cathcart and El-Jahel [1998] LS 10 () a 1970 1990 O/N 6.890 X, s, w, r Spread (6) JGB 11 log( Pt ( X, r, T) / D( r, T )) log(1 - wq( X, r, T )) Spread = - = - (6) T T w, r 10 11 -rt -( r+ Spread ) T (6) D( r, T ) = e P ( X, r, T) = e (5) t
w 0.9 0.1 1 r 0 13 (6) ( X, s ) T Q( X, r, T ) 14 X, s LS V r s Z 1 dv = rvdt + svdz 1 V K X = V / K 1 1 [1999] 13 JLT 14 Q( X, r, T ) 00 n n=100
X s s K X (6) s X X s [ 4] s =0.5 bps 3000 500 000 1500 1000 500 0 0 1 3 4 5 6 7 X=4 X=3 X=
[ 5] X =3 bps 3000 500 000 1500 1000 500 0 0 1 3 4 5 6 7 =0.3 =0.5 =0.7 X s humped-shape t ( t, t + 1] 15 16 LS 15 humped-shape 16 Sarig and Warga [1989]
LS X, s 17 JGB LS LS 8 AAA 18 K V - K 1 = 1- X V X BB 36.7% A 1.1%AAA 17.8% 36.7% 17 18 [1998]
X [ 6] BB bps 700 600 500 400 300 00 100 0 0 1 3 4 5 6 7 X=1.580=0.3 [ 7] A 160 140 10 100 80 60 40 0 0 0 1 3 4 5 6 7 X=1.67=0.108
[ 8] AAA 50 45 40 35 30 5 0 15 10 5 0 0 1 3 4 5 6 7 X=1.16=0.081 [ 9] 30 5 0 15 10 5 0 AAA AA A BBB BB B
[ 10] B 4.5 4 3.5 3.5 1.5 1 0.5 0 AAA AA A BBB BB
JLT Kijima and Komoribayashi [1998] JLT [1999] (structural model) (reduced form model) LS JLT JLT [1999] 19 [ 11] LS AAA AA A BBB BB B JLT 0.60% 0.79% 1.3%.31% 6.55% 31.48% LS 0.0% 0.5% 0.49% 0.87% 3.74% 7.58% JLT 19 [1999] 1998 10 9 YTM 0.1 [1999]
JLT 0 LS LS JLT LS LS JLT LS 0 JLT
JLT LS YTM JGB
Cathcart, L. and L. El-Jahel, Valuation of Defaultable Bonds. Journal of Fixed Income, June 1998, pp.65-78. Duffie, D. and K. J. Singleton, Modeling Term Structures of Defaultable Bonds. Working Paper, Graduate School of Business, Stanford University, 1998. Kijima, M. and K. Komoribayashi, A Markov Chain Model for Valuing Credit Risk Derivatives. Journal of Derivatives, Fall 1998, pp.97-108. Jarrow, R. A., D. Lando and S. M. Turnbull, A Markov Model for the Term Structure of Credit Risk Spread. Review of Financial Studies 10 (), 1997, pp.481-53. Longstaff, F. A. and E. S. Schwartz, A Simple Approach to Valuing Risky Fixed and Floating Rate Debt. Journal of Finance 50 (3), 1995, pp.789-819. Sarig, O. and A. Warga, Some empirical estimates of the risk structure of interest rates. Journal of Finance 44, 1989, pp.1351-60. IMES Discussion Paper No.99-J-18 1999 6 Libor IMES Discussion Paper No.98-J-10 1998 7 18 1 1999 3