IMES DISCUSSION PAPER SERIES Discussion Paper No. 99-J- 9 -J-19 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN

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1 IMES DISCUSSION PAPER SERIES Discussion Paper No. 99-J- 9 -J-19 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN

3 IMES Discussion Paper Series 99-J- 9 -J * * [1999] *( akira.ieda@boj.or.jp, toshinao.yoshiba@boj.or.jp)

4 . JLT

5 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

6 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]

7 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

8 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

9 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

10 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 LS () () dr = ( a - br) dt + hdz () O/N AA A 9 B 1 B

11 a, b, h Libor r [ 3] a b h r Cathcart and El-Jahel [1998] LS 10 () a O/N 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 rt -( r+ Spread ) T (6) D( r, T ) = e P ( X, r, T) = e (5) t

12 w 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

13 X s s K X (6) s X X s [ 4] s =0.5 bps X=4 X=3 X=

14 [ 5] X =3 bps =0.3 =0.5 =0.7 X s humped-shape t ( t, t + 1] LS 15 humped-shape 16 Sarig and Warga [1989]

15 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% [1998]

16 X [ 6] BB bps X=1.580=0.3 [ 7] A X=1.67=0.108

17 [ 8] AAA X=1.16=0.081 [ 9] AAA AA A BBB BB B

18 [ 10] B AAA AA A BBB BB

19 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] YTM 0.1 [1999]

20 JLT 0 LS LS JLT LS LS JLT LS 0 JLT

21 JLT LS YTM JGB

22 Cathcart, L. and L. El-Jahel, Valuation of Defaultable Bonds. Journal of Fixed Income, June 1998, pp Duffie, D. and K. J. Singleton, Modeling Term Structures of Defaultable Bonds. Working Paper, Graduate School of Business, Stanford University, Kijima, M. and K. Komoribayashi, A Markov Chain Model for Valuing Credit Risk Derivatives. Journal of Derivatives, Fall 1998, pp 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 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 Sarig, O. and A. Warga, Some empirical estimates of the risk structure of interest rates. Journal of Finance 44, 1989, pp IMES Discussion Paper No.99-J Libor IMES Discussion Paper No.98-J

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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 CDO 2010 5 18 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 1 1.1 2007 2008 9 15 ( ) CDO CDO 80 E-mail:taiji.ohka@gmail.com