CDOのプライシング・モデルとそれを用いたCDOの特性等の考察： CDOの商品性、国内市場の概説とともに

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1 CDO CDO CDO CDO CDO CDO

2 CDOcollateralized debt obligation CDO CDO CDO CDO CDOCDO CDO CDO CDO CDOCDO CDO CDO CDO CDO CDO CDOABSasset backed securities CBOcollateralized bond obligation CLOcollateralized loan obligation CDOCDO

3 CDO ABSCDO CDO CDO CDO CDO CDO

4 CDO CDO CDO CDO SPVspecial purpose vehicle SPVCDOCDO BIS CDO CDO CDOCDO SPV CDO Libor + CDO CDO SPVCDO CDO CDO SPV CDO SPV Fabozzi and Goodman CDO

5 CDO CDOCDO CDO CDOCDOCDO CDO SPV CDScredit default swap TRStotal return swap CDS CDOCDS CDS CDO CDOSPV bankruptcyfailure to payrestructuring ISDAhttp:// > CDOCDS

6 CDOCDO CDO CDS CDO CDO CDO CDO SPV CDSSPV SPV CDS CDS

7 CDOCDO CDO CDO CDOCDO CBO CDO

8 CDO CDO CDO CDO CDO CDO CDO CDO CDS CDO CDO CDS CDS CDOCLO CLO CLO CDO

9 CDO CDO CDO CDS CDSCDO CDO CDO CDO CDS CDO BIS CDO BISGoodman CDO CDSCDO CDS

10 CDO CDS OECD CDO CDO CDO CDO CDO

11 CDO p i p j i j p i j a ij p ( p ) ( p i j ) i j = a ij u exp a ij uv + v ( a ij ) dudv uv (.) i j d ij d pi j pi p j ij = p ( p ) p ( p ) i i j j p i p j i j d ij d ij

12 CDO CDO J. P. Morgan & Co. Nagpal and Bahar CDO

13 CDO Cifuentes and O ConnorFinger FingerLi Duffie and Gârleanu Cifuentes and O Connorbinomial expansion technique modelcifuentes and O Connor CDO CDO N X i i i pn j N j j N j C p ( p)

14 Cifuentes and O Connordiversity score MM N i (i = N ) i CDOTL i R i n i L i (T) = ( R i )n i {i T} L (T) N i =L i (T) i T p i N E [ L( T)] = ( R ) n p i = i i i V [L (T)]V [L i (T)] = ( R i ) n i p i ( p i ) N V [ L( T)] = V[ L i( T)] + i = N i j i j d ij V [ L ( T )] V[ L i j ( T)] d ij i j d ij i j p i j d ij = i j p ( p ) i p p pj i i p ( p ) j j. MM M N R p E[ L ( T)] = ( R ) NMp V [ L( T)] = ( R) N M p ( p). {i T} i T

15 N = N i = n i /M R = N i = n i R i /( NM) N ( Ri) ni p i p = = ( R ) NM i M E[ L( T)] ( R ) ni i = = V[ L( T)] N E[ L( T )] p i = p j d ij = N M = + ( N ) alternative diversity score Finger CDO i {X i (t)} t X i (t) i (t) X i (t) Backman and O Connor

16 (.)( i (t)) CDO i p i () ( i ())= p i () i () i j d ij Z i () iz i ()< i () CDO CDOCDO i p i () ( i ())= p i () i () i j d ij Z i () i Z i ()< i () Z i () Z i ()Z i () q i () i ()= (q i ()) i Z i ()< i () i t p i (t)(t )t q i (t) (p i (t ) p i (t ))/( p i (t ))

17 CDO copula X X n F F n F [ F ( X ) F ( x ) F ( X ) F ( x )] F( x xn ) = P n n n n = C F ( x ) F n( x ( n )) C F (X ) F n (X n ) 0 C0 F(t)= P( t) U = F (U) F F n U U n P t t ] = P[ U F ( t ) U F ( t )] C( F t ) F n t )) [ n n n n n = ( ( n = F (U ) n = F n (U n ) Nelsen Li

18 normal (n) (. ;) ( n) n n n n C( u u ) = P( U u U u ) = ( ( u ) ( u ); ) (.) i j d ij Z (i) U i = (Z (i) ). = F i ( i) i ( Ui ) = Fi ( ( Z )). n i j t () ( (F i (t)) (F j (t)) ;) ij Merton a ij ij = a ijschmidt and Ward

19 i p i () p i () p i (T) k = T i (k)= (p i (k)) i j d ij Z(i) i (k )< Z (i) i (k) i k Gumbel CDO CDO C C( u u u / p u p ) = exp( {( lnu ) + + ( ln ) } ) n > [0 ] VW F(x) = e x w0 u w = F (u) = ln ( u)

20 sin(( ) V / ) = W sin( V / ) sin( V ). 0 U U n k = n X X n X k k ln( U k)} = F (exp[ { ]). Marshall and Olkin Marshall and Olkin LT (s)= E [e s ] F( x x x ) = E [ H ( x ) H ( x ) H n = LT ( LT ( F ( x )) + + LT ( x n n ) ( F ( x n ] n ))) LT(s) = exp ( s / )Kanter LT(s) exp ( s / ) / H ( x ) k k = exp( LT ( F ( x k k ))) for k = n H k (X k ) 0 X k XY (X Y ) cov( X Y) ( X Y) = V[ X ] V[ Y] cov(x Y ) XYV[.] Joe

21 XY XY Duffie and Gârleanudefault intensity t (t) t t Embrechts McNeil and Straumann

22 (t) ( σ l )basic affine W J(t) t W l t t + s (s) (s)e t t CDO Duffie and Gârleanu Duffie and GârleanuDuffie and Singleton. t t t t t P t + < ) ( ) ( [ ] ) ( ) ( ) ( ) ( ) ( t J t dw t dt t t d + + = ) ( ) ( ) ( exp ) ( t s s s t t u t t e du E s t P + + = = + > s c d c e d c d c b d c s s b ln ) ( ) ( = s l c l d c e d c d c b d c a l s b ln ) ( s b s b e d c e s ) ( + = + = b + + = = d + c d a = b b = c c = c d d + =. c

23 X( X σ l X )Y( Y σ l Y )XY X+Y( σ l ) = X + Y l = l X + l Y N N = X + X i C i X C NX i i X C X i ( C σ l C )( i σ l i )X X N X C i ( σ l ) = C + i l = l C + l i t i j jump lc = l lc = l + l C i i j jump = 0 i j X C Duffie and Gârleanu jump i jump ij X C X i X j J C J i J j i j J i = J C +J i J j = J C +J i cov[ J J ] i j jump ( J J ) i = j V[ J ] V [ J ] i j jump( J J ) j. i = l C t l = C l t l t l Schönbucher

24 Duffie and Gârleanu ( σ l ) TP 0T f T ( ) + ( T) (0) 0 T ) 0 P T = ( T) e + f ( u) ( u du (T )T ( u) = d du P( > u) = e [ ( u) + ( u) (0) ] ( u ) + ( u) (0) ( u) ( u) = ( u) + l ( u) = ( u) + ( u) ( u) CDO i i X ( t) = X ( t t) + ( X ( t t)) t + X ( t t) t J ( t) C C C C C C + X ( t ) = X ( t t) + ( X ( t t)) t + X ( t t) t J ( t ). i i i i i i + i C C i N (0 ) mk Jk( t) = 0 pr = l t pr = l t m k (k = C i ) i X C X i t t i (t) t N + N + [0 ] U0 m k = ln ( U)

25 CDO CDO CDO CDO CDO p0 p/00

26 CDO CDO CDO CDO

27 R&I Nagpal and Bahar

28 CDO

29 d ij E[L E ]p 7 00 k 00 k 9k k 00 k C p p C E] = 7 + k p p k= 0 00 ( ) k = 8 00 ( ) 70 E[ L

30 CDO k 9 k/70 k k MR qe[l E ] M ) k M k k( R E[ L ] = C p ( p) min E M k 0 qm k= ME[L E ]E[L E ]M M > 0 E [L E (M+)] E[L E (M)] 0 Rq pe[l E ]p E[L E (M)]E[L E (M+)] E[L E (M)] E[L E (M)]M E[L E (M+)] E[L E (M)] 0 Mp E[L E (M+)] E[L E (M)] MM

31 E[L E (M)] E[L E (M + )] E[L E (M)]

34 p i 55 C i p i ( p ) 55 i CDO CDS CDO Backman and O Connor

35 CDOCDO

36 CDS CDO CDO CDOCDO CDO CDO CDS

37 CDO CDO CDOCDO CDOCDO

38 CDO CDO

39 CDO CDO CDO CDO Howard Lee and Mancini CDO

40 CDO overcollateralization testoccdo CDO CDO CDOCDO minimum net worth testmnwcdo

41 CDO CDO Moody s Japan K.K CDO Backman A. and G. O Connor Rating Cash Flow Transactions Backed by Corporate Debt 995 Update Moody s Investors Service 995. Cifuentes A. and G. O Connor The Binomial Expansion Method Applied to CBO/CLO Analysis Moody s Investors Service 996. Duffie D. and N. Gârleanu Risk and Valuation of Collateralized Debt Obligations Financial Analyst Journal 57 () 00 pp and K. J. Singleton Credit Risk Pricing Measurement and Management Princeton University Press 003. Embrechts P. A. McNeil and D. Straumann Correlation: Pitfalls and Alternatives Risk (5) May 999 pp Fabozzi F. J. and L. S. Goodman Investing in Collateralized Debt Obligations John Wiley & Sons 00. Finger C. C. A comparison of stochastic default rate models Working Paper The RiskMetrics Group 000. Goodman L. S. Synthetic CDOs: An Introduction The Journal of Derivatives 9 (3) 00 pp Howard D. J. Lee and M. J. Mancini Market Value CBO/CLO Rating Criteria Fitch Ratings 999. Joe H. Multivariate Models and Dependence Concepts Chapman & Hall 997. J. P. Morgan & Co. CreditMetrics TM Technical Document April 997. Kanter M. Stable Densities Under Change of Scale and Total Variation Inequalities The Annals of Probability 3 (4) 975 pp

42 Li D. X. On Default Correlation: A Copula Function Approach The Journal of Fixed Income 9 (4) 000 pp Marshall A. W. and I. Olkin Families of Multivariate Distributions Journal of the American Statistical Association 83 (403) 988 pp Merton R. On the pricing of corporate debt: the risk structure of interest rates The Journal of Finance pp Nagpal K. and R. Bahar Measuring default correlation Risk 4 (3) March 00 pp Nelsen R. B. An Introduction to Copulas Springer New York 999. Schönbucher P. Term Structure Modelling of Defaultable Bonds Review of Derivatives Research (/3) 998 pp Schmidt W. and I. Ward Pricing default baskets Risk 5 () January 00 pp. -4.

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

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