住宅ローン債権担保証券のプライシング手法について：期限前償還リスクを持つ金融商品の価格の算出

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1 MBS

2 Residential Mortgage-Backed SecuritiesRMBS Mortgage-Backed SecuritieMBS mortgageasset-backed SecuritiesABS Commercial Mortgage-Backed SecuritiesCMBSRMBSMBS MBSMBS MBSMBS MBS MBS MBS MBS MBSMBS MBS MBS MBS MBS MBS MBS MBSMBS

3 MBS ABS MBS MBS MBS MBS MBS MBSGNMAGovernment National Mortgage Association FHLMCFederal Home Loan Mortgage CorporationFNMAFederal National Mortgage AssociationMBSMBS Pass-Through CMOCollateralized Mortgage Obligation Pay-Through

4 MBS MBS SPCSpecial Purpose CompanyMBS MBS

5 MBS MBS ALMAsset Liability Management MBS MBS MBS MBS MBS MBS MBS MBS MBSNOMURA-BPI

6 CDO Collateralized Debt Obligation MBS ALM MBS MBS MBS MBSMBS CMOCollateralized Mortgage Obligation CMOABC A AB C SMBSMBSStripped Mortgage-Backed Securities

7 IOInterest Only POPrincipal Only IOPO IOPO MBSABS MBS

8 MBS MBS MBS AMBS MBS MBS MBS MBSMBS

9 MBS MBS MBS MBS MBS MBS MBS MBSMBS MBS mt c t i = i / m (i = 0,1,, mt ) AM(t i ) MBS

10 c A = M(t i ) M(t i+1 ) + M(t i ), m t 0 = 0 M ( 0) = = A 1 + c / m A 1 + c / m M ( t 1) c / m A + ( 1 + c / m ) + A + ( 1 + c / m ) mt = A 1 1/ ( 1 + c / m ) c / m mt, c/m(1+c/m) mt A = M(0), (1+c/m) mt 1 t i (1+c/m) mt (1+c/m) mt i M (t i ) = M(0), (1+c/m) mt 1 t i P(t i )

11 c/m(1+c/m) mt i1 P(t i ) = M(t i1 ) M(t i ) = M(0), (1+c/m) mt 1 t i I (t i ) c c (1+c/m) mt (1+c/m) mt i1 I (t i ) = M(t i1 ) = M(0). m m (1+c/m) mt 1 (t i1, t i ] SMM(t i ) t i S(t i ) S(t i ) i S (t i )=Π{1 SMM(t j )}, j =1 S(t i1 ) S(t i ) SMM(t i ) =. S(t i1 ) M (t i )S(t i ) M (t i )=M(t i )S(t i ). A (t i ) t i1 M (t i1 ) T t i1 c/m(1+c/m) m(tt i1 ) A (t i ) = M (t i1 ) = AS (t i1 ), (1+c/m) m(tt i1 ) 1 I (t i ) t i1 M (t i1 ) c I (t i ) = M (t i1 ) = I (t i )S(t i1 ), m P (t i )A (t i ) SMMSingle Monthly Mortality SMMCPRConditional Prepayment Rate

12 I (t i ) P (t i ) = A (t i ) I (t i ) = P(t i )S(t i1 ). t i t i1 M (t i1 )t i P (t i ) (t i1, t i ] PR(t i ) SMM(t i ) PR(t i ) = {M (t i 1 ) P (t i )} SMM(t i ). MBSCF(t i ) P (t i )I (t i )PR(t i ) CF (t i ) = P (t i ) + I (t i ) + PR(t i ) = P(t i )S(t i 1 ) + I (t i )S(t i1 ) + {M(t i 1 ) S(t i1 ) P(t i )S(t i1 )}SMM(t i ) = {M(t i 1 ) + I (t i )} S(t i1 ) M(t i ) S(t i ). FabozziKariya and Kobayashi

13 A ma m t cm(0) A =, 1 exp {ct} A 1 exp { c (T t )} M(t) = (1 exp { c (T t )}) = M (0). c 1 exp { ct } CF(t i )= A (t i ) + PR (t i ), A (t i )= AS (t i1 ), PR(t i ) = {M (t i 1 ) S(t i1 ) P(t i )S(t i1 )}SMM(t i ) ={M (t i 1 ) P(t i )} {S(t i1 ) S(t i )} = M (t i ){S(t i ) S(t i1 )}, t t i t i 1 0m dcf (t )=A S(t ) dt M (t ) ds (t ). MBS S(t) S(t)

14 S(t) MBS t ( 0)t + t f (t) t = Pr (t < t + t ) F ( t ) = Pr ( t ) = f ( s) ds, S ( t ) = Pr ( > t ) = f ( s) ds. t 0 t Pr (t < t + t > t ) h (t ) lim. t 0 t d h (t ) = ln S (t), dt t H(t ) h(s)ds, 0

15 S(t )= exp { H(t )}, t S ( t i ) exp = Π i j = 0 Π i j = 0 i j = 0 h ( t j ) t exp{ h ( t j ) t } { 1 h ( t j ) t }. SMM(t i ) z (t)t h(t ) h (t, z (t)) = h(t ) exp { z (t) }. h(t ) t Jegadeesh and Ju Sugimura

16 Schwartz and TorousSugimura h(t )= (t ) 1, (t ) 1 h(t ) =, 1+ (t ) h (t ) = t ln t ln t. (x)φ(x) z 1 (t ) ctr(t ) z 1 (t )= c R(t ),

17 z 1 (t )= c /R(t ), 1 R(t ) MBS Schwartz and Torous z (t ) z (t ) = ln AO (t ) AO (t ), AO (t )t AO(t ) AO (t ) McConnell and SinghStanton McConnell and Singh t r(t ) D (t, r(t)) F (t ) RF D (t, r(t)) (1+ RF ) F (t ),

18 r (t) r (t) (t) r (t) r (t) (t)+ if r (t) r (t), h (t) = (t) if r (t)> r (t). (t) MBS N S n (t )( n = 1,,, N )n n S (t) N = 1 S ( t ) = N, 1 n S n ( t ) n =. n= 1 n

19 MBS MBS MBS r (t) h(t) h(t) = h (t, r(t)), h(0) 0. {r (t), 0 t T } dr (t) =(r (t), t ) dt + (r (t), t ) dw(t). (r (t), t ) (r (t), t ) r (t)t W (t) MBSV V = E mt exp t i r ( s) ds CF ( t i =1 0 i ), V = E T exp t r( s) ds dcf ( t) 0, 0 MBS E[.] MBS

20 i mt 1 { M( t i 1 ) + I ( t i )} exp r ( t ) j t S ( t i 1) i =1 j = 0 mt i =1 i 1 M( t i ) exp r ( t j ) t j = 0 S ( t i ), Schwartz and TorousMcConnell and SinghStanton MBS CIRCox, Ingersoll and Ross dr (t) = a( r r (t)) dt + r (t) dw(t), r (0) 0, MBS t MBSV (t)l (t) L ( t ) = t exp r ( u) du V (t) + 0 t exp s du 0 r ( u) CF (s) ds 0 = t exp r ( u) du V (t 0 ) + t exp 0 r ( u) du { AS (s) + M ( s) h ( s ) S( s) } ds 0. L (t)mbsv (t)t CF (s) s

21 L (t) MBSV (t)rst V (t) V(r, S, t )S (t) ds (t) = h(t) S (t) dt, t 0, MBSV (r, S, t ) dv(t) = (r, S, t )dt + (r, S, t ) dw(t), = a ( r r (t)) V h( t) S( t ) V V r ( t) V r S t r = r( t) V. r, t exp Z ( t ) r ( u) du, t 0, 0 dz(t) = r (t) Z (t) dt, dl( t ) = V( t ) dz ( t ) + Z ( t ) dv( t ) + d Z, V ( t ) + Z ( t) { AS ( t ) + M( t) h ( t ) S( t) } dt = Z ( t )( r ( t ) V ( t ) + a ( r r ( t) ) V t V r h ( t) S( t ) V S ) dt r ( t) V + AS ( t) + M( t ) h ( t ) S( t) r + Z ( t ) r ( t) V r dw ( t). L (t) Z, V ZV quadratic covariation Musiela and RutkowskiAppendices B

22 V V r (t)v (t)+ a ( r r (t)) h(t) S (t) r S V 1 V + + r (t) + S (t)+ M (t) h(t) S (t) = 0, t r MBS V (T ) = 0 MBS MBS t t i t i 1 = 1/mN t T t n k(n, k) t n k (n, k)r (n, k)(n, k)(n + 1, l ) p (n; k, l )t n J n i t t i AN t T Schwartz and Torous

23 n t k A(n, k)n t ki tv (n, k; i ) A N A(n, k) =A v (n, k; i ). i =n +1 (n, k)h (n, k) t 1 h (n, k) t(n, k) C (n, k) C ( n, k ) = h (n, k ) t f (n, k ) = e r (n, k ) t (1h (n, k ) t) p(n; k, l ) C (n +1, l ). J l n +1 f (n, k)(n, k) f (n, k)= A(n, k) M(t n ), C (N, k) = 0 C ( 0, 0) V = A(0, 0) C (0, 0). h (n, k ) = 1 t if 0 otherwise, f (n, k ) e r (n, k ) t p(n; k, l ) C (n +1, l ), J l n +1 (n, k)n t v (n, k; N ) = E N j = 1 exp n r ( tj ) t r ( t n) = r ( n, k ) = e r (n, k ) t j N = 1 E exp n +1 r ( tj ) t r ( t n) = r ( n, k ) = e r (n, k ) t l J n +1 p (n; k, l ) v (n +1, l; N ), k v (N, k; N )=1

24 C (n, k ) = max f (n, k ), e r (n, k ) t p (n; k, l ) C (n +1, l ), J l n +1 MBS X (0) X ( 0) = E exp t i r ( s) ds X (t ) = v, t t ( 0 ) 0 i i E i [ X ( t i )], v (0, t i ) t i E t i [.] Pr t i MBS mt { i i } i =1 t V = M( t 1 ) + I ( t ) v ( 0, t ) E i [ S ( t ] i i ) 1 mt i =1 t M( t i ) v ( 0, ti ) E i [ S( t i )], v (0, t i )E t i [ S (t i )]E t i [ S (t i 1 )] MBS IO =E mt i =1 exp t i r ( s) ds I (t ) = i 0 mt i =1 t I (t i ) v ( 0, ti ) E i [ S ( t i 1) ],

25 PO =E mt i =1 t i exp r ( s) ds { P (t i ) + PR(t i )} 0 = mt t M( t i 1 ) v( 0, t ) E i [ S ( t i ) ] + i =1 i 1 mt i =1 M( t i ) v ( 0 t, t ) i E i [ S ( t ], i ) MBSIOPO Hull and White dr (t) = ((t) ar (t)) dt + dw (t), r (0) 0. Pr t i dr (t) = {(t) +b t i (t) ar (t)} dt + dw t i (t), 0 t ti. W t i (t) Pr t i 1e a(t i t ) b t i (t) =, a dx (t) = ax (t) dt + dw t i (t), X (0) = 0, {X (t), 0 t T }X (t) t i (t) r (t) = X (t)+ t i (t), dr (t) = { t i (t)+ a t i (t) ar (t)} dt + dw t i (t),

26 t i (t)+ a t i (t) = (t)+b t i (t), t i (0) = r (0), t i (t) t i ( t ) = at e ( ( s) + b ( s) t e as t i ) ds + r ( 0), 0 r (n, k) x(n, k) r (n, k) = x (n, k)+ t i (t n ). (t) t i (t)= (t) + t i (t), t i ( t ) = at e b ( s ) t e as t i ds 0 = a e a( t i t ) e a( t i + t ) + e a t 1. t i (t) (t) t i (t) MBS (n, k) Q(n, k) (n+1, l ) Q(n+1, l )=p(n;k, l )exp{h(n+1, l ) t}q(n, k), k J n

27 Q(0, 0)=1Q(n+1, l ) (n, k) (n+1, l ) E t i [ S (ti )] = Q(i, l), l J i E t i [ S (ti 1 )] = Q(i 1, k). k J i1 MBSIOPO MBS MBS MBS MBS

28 MBS MBS MBSV V = E = mt i =1 mt i =1 t i exp r ( s) 0 ds CF ( ti) {M( t i 1 ) + I (t i )} E exp r ( s) ds 0 t i 1 S (t i ) mt M(t i )E exp ds r ( s) 0 i =1 t i S (t i ), M(t i ) I (t i ) t i E exp r ( s) ds S ( t 0 i 1), t i E exp r ( s) ds S(t 0 i ), MBS t i E exp r ( s) ds S (t i 1) 0 E exp t i r ( s) ds S ( t i ) 0 = v ( 0, ti ) S( t i 1), = v ( 0, ti ) S( t i ), v (0, t i )S (t i ) MBS PSAS (t i ) PSAPSAPublic Securities AssociationMBS

29 CPR(t i ) PSA t i i i 6% if i CPR (t i ) = 6% if i > 30 = 6% min 1, i, 30 SMM (t i ) = 1 (1 CPR (t i )) 1/1 S ( t ) = i Π i j = 1 {1 SMM ( t ) j j } = 1 6% min 1,. j = Π i MBS MBS t i MBS 1

30 NakamuraMBS MBS MBS h(t) = e (Lr(t)). L {r (t), 0 t T } Vasicek dr (t) = a( r r (t)) dt + dw(t), r (0) 0. a r W(t) E exp r ( s) ds 0 t i 1 S(t i ) t = v ( 0, t ) E i [ S( t i )], i 1 E exp t i r ( s) ds S( t i ) 0 t = v ( 0, t ) E i [ S( t i )]. i v (0, t i )MBS E t i [ S (t i 1 )]E t i [ S (t i )] Pr t i {r (t), 0 t T } Cox

31 dr (t) = a((t)r (t)) dt + dw t i (t), 0 t ti, W t i (t) a (t) = r ( 1e a(t i t ) ), d lnh(t) = dr (t) = a ( (t) r(t) ) dt dw (t) t i lnh(t) = a (t) L + dt dw t i (t) t = a ( ( L (t)) lnh(t) ) dt dw i (t). (t) a(l (t)), d ln h(t) = ((t)a ln h(t)) dt + dw t i (t), h (0) 0, Black and Karasinski t E i [ S ( t i )] 1 = E t i exp t i 1 0 h( s) ds t E i [ S( t i )] E exp = t i ds h( s), 0 t i, MBS MBS MBS Brigo and Mercurio

32 R = 0.88 CPRSMM MBS

33 h (t) = (L r (t)), h (0) 0. L{r (t), 0 t T } dr (t) = a ( r r (t)) dt + dw (t), r (0) 0. a r W(t) dh(t) = dr (t) = a((t)r (t))dt dw t i (t) = a ((L(t))h(t))dt dw t i (t), dh(t) = ((t)ah(t)) dt + dw t i (t), Hull and White E t i [ S (t i )]E t i [ S (t i 1 )] E [ S (t)] (x) (1e ax )/a 1e a(t t ) b T (t) = = (T t), a a ( t) T = b T t T T = a ( L ) t (s) bt ( s) ds + a ( T ) a ( s) ( s) ds + 1 tt b T ( s) ds t T bt = r L (( T t) T t a + e + a ( T t) ( T t) ( )) ( T t ) + ( s) ds ( ( T t)) (( T t)),

34 E exp t E [ S( t) ]= h( s) ds 0 = exp{ a ( ) + b t ( 0) h ( 0) } t 0 = exp r L ( t ( t )) + e a a ( t ) a ( t) (t) + a (t) t ( t ) + (t) ( t) (L r ( 0)) = exp r L a ( t ) a 1 + ( e ) t r ( 0 ) r a ( t) a ( t ) + a e (t), t E [ S( t i) ] = exp r L a 1 + t r ( 0 ) r + i i ( ) ( t ) a i + a 1 (t i ), t E [ S ( t i ) 1 ] = exp r i L a 1 ti 1 + r ( 0 ) r + a e a (t i t i1 ) (1 + ) ( t i 1) ( a e t ) i 1, + a (t i t i1 ) MBS MBS MBS MBS

35 MBS DuffieKijima and Muromachi MBS h (t) = (L r (t)) + g (t), h (0) 0. Lg(t) {r (t), 0 t T } dr (t) = a( r r(t)) dt + dw 0 (t), r (0) 0, {g (t), 0 t T } dg(t) = b ( gg(t)) dt + dw 1 (t), a r b gw 0 (t)w 1 (t) dw 0 (t)dw 1 (t) = dt PSA g(t) SMM/ t +

36 t i E exp r ( s) ds S( t = E 0 i ) exp r ( s) ds 0 = E exp t i t i + 0 t i exp h( s) ds 0 ) ((1 r ( s) g( s)+ L) ds, t i ti ) E exp r ( s) ds S( 1 = E exp t i 0 ds r ( s) 0 exp t i 1 0 h( s) ds t i = E exp t r ( s) ds i1 0 ( r( s) + g( s)+ L) ds 0, MBS {r (t), t 0}OUOrnstein=Uhlenbeck r ( t) = r + ( r ( 0) r ) e at + 0 t e a (t s) dw 0 ( s), H( ) r ( t ) dt = r 0 dt + ( r ( 0) r ) e at 0 dt + 0 t 0 0 e a (t s) (x) (1e ax )/a H( )= t 0 0 e a (t s) dw 0 ( s) dt, 1 dw 0 ( s) dt a (t s) e = e dt dw, s 0 ( s) = 0 a ( s) dw 0 a 0 ( s) r + ( r ( 0) r ) ( ) ( s) + 0 dw 0 ( s), H() H () E [H()] = r + (r(0) r ) (),

37 ( ) H V [H( )] = ( s) ds = ( e a ( s) ) 0 a 1 ds 0 = ( 1 ) + a ( ). {g (t), t 0}OU g ( t) = g + bt ( g( 0) g t ) + 0 b (t s) e e dw 1 ( s), (x) (1e bx )/b G( ) g( t ) dt = g + g( 0) g 0 ( ) ( ) + ( s) dw 1 ( s), 0 G() ( ) E [ G ( )] = g + ( g( 0) g ) ( ), G V [G G ( ) ( )] = ( 1 b ) + ( ). H()G()cov (H(), G()) cov(h( ), G( )) = E ( s) dw0 ( s) 0 ( s) dw1 ( s) 0 = ab 0 b ( s ) a ( s ) ( 1 e ) ( 1 e ) ds = ab = ab 1 e a a ( ) ( ) + 1 e b b + 1 e a+b 1 e a+b (a+b) (a+b), H()H()cov(H(), H()) cov(h( ), H( )) = E ( s) dw0 ( s) 0 ( t ) dw0 ( t ) 0 min{,} a ( s ) = 0 ( 1 e ) ( 1 e ) ds a a ( s ) = a ( ) + a (1 e ) ( ) + e a ( ) ( ).

38 H()G()cov(H(), G()) cov(h( ), G ( )) = E ( s) dw0 ( s) 0 ( t ) dw1 ( t ) 0 t i = +. ab e a ( ) e e a b ( ) ( ) a ( ) a+b Y ( t i ) ((1 ) r( s) + g( s)+ L ) ds 0 t i = ( 1 ) r 0 t i ( s)ds + g( s)ds + L 0 Y(t i ) Y(t i ) t i = ( 1 ) H( t i ) + G( t i ) + L t i, Y (t i ) E [Y(t i )] = (1) H (t i )+ G (t i )+ Lt i, Y (t i ) V [Y(t i )] = V [(1)H(t i )+ G(t i )] = (1) H (t i )+ G (t i )+(1 )cov(h(t i ), G(t i )), cov(h(t i ), G(t i )) E exp t i ds r ( s) S( t i ) 0 E [ e ] = Y ( ti ) = exp t i t 1 Z ( t i ) r ( s) i ds + ( r ( s) + g( s) + ) 0 L ds 0 Y Y ( t ( t i ) + i ). = H ( t i ) H t i ) ( 1 + G 1 ( t i ) + L t i 1, Z(t i )Z(t i ) XN(, )m X () E [e X ] m X () = exp { + },

39 Z (t i ) E [Z(t i )] = H (t i ) H (t i 1 )+ G (t i 1 )+ Lt i 1, Z (t i ) V [Z(t i )] = V [H(t i ) H(t i 1 )+ G(t i 1 )] = H (t i )+ H (t i 1 )+ G (t i 1 )cov(h(t i ), H(t i 1 )) +cov(h(t i ), G(t i 1 )) cov(h(t i 1 ), G(t i 1 )), cov (H (t i ), H(t i 1 ))cov(h (t i ), G (t i 1 )) cov (H(t i 1 ), G(t i 1 )) E exp t i ds r ( s) S ( t i 1) 0 = Z ( ti ) E [ e ] = exp ( t ( t i ) + i ), MBS MBS g( t)= 0 Z Z MBS

40 MBS MBS {r (t), t 0} dr (t) = a( r r(t)) dt + dw(t), t 0, MBS 1 (t) h(t) = e (R r(t)), 1+(t) MBS MBS

41 a R r r (0) t

42 MBS MBSMBS MBS MBSIOPO ED V y V y ED = 100. V 0 y V 0 MBS y V y (V y ) y ( y )MBS y =0.1% MBS = 0 a R r r (0)

43 MBS MBS = 0 = 5 = 0

44 IO = 0IO IO POPO = 0

45 MBSPSA MBS g (t) PSA MBS PSAMBS =.0MBS b L g g(0)

46 r(t)g (t) MBS MBS PSA

47 MBS MBS MBSIOPO PSA MBS PSA MBS MBS

48 MBSMBS MBS

49 MBS {r (t), t 0} dr (t) = (r(t), t) dt + (r(t), t)dw(t), t 0, W (t ) (r, t)(r, t) (r, t) = 1 (t) + (t)r, (r, t) = 1 (t) + (t)r, affine modela T (T ) = b T (T) = 0 ( t ) b ( t ) = ( t ) bt ( t ) T ( t ) + 1, bt at 1 ( t ) b ( t ) = 1 ( t ) bt ( t ) T ( t ), a T (t )b T (t )v (t, T ) v (t, T ) = exp {a T (t) + b T (t) r (t)}, Y (t, T ) b T (t) a T (t) Y(t, T ) = r (t), T t T t

50 {r (t), t 0} dr (t) = a( r r(t)) dt + dw (t), t 0, (r, t) = a r ar, (r, t) =, at bt ( t ) = ( bt ( t ) + T t ) r ( t ) = 1 a (T t) e a a r (t) 4a b T ( t ),, {r (t), t 0} dr (t) = a( r r(t)) dt + r(t) dw (t), t 0, CIR (r, t) = a r ar, (r, t) = r, at ( t ) = a r ln e (a + ) (T t) / ( a+ )( e (T t) 1) +, bt ( t ) = ( e (T t) 1) ( a+ )( e (T t) 1 ) +, = a + CIRr (t)

51 {r (t), t 0} dr (t) = ((t) ar(t)) dt + (t) dw (t), t 0, (r, t) = (t) ar, (r, t) = (t), at T t ( t ) = ( s) bt ( s)ds + 1 T t ( s)b T ( s) ds, bt ( t ) = 1 a (T t) e a, (t) (t) X (0) X ( 0) = E exp T ds r ( s) X(T ) = v(0, T ) E T [X (T )], 0 v (0, T ) T E T [.] Pr T X (T ) T tv T (t, ) v (t, ) v T (t, ) = = exp {a (t) a T (t) + (b (t) b T (t)) r (t)}, v (t, T )

52 d ln v T (t, ) = ( b ( t ) bt ( t ) ( r, t ) b ( t ) bt ( t ) ( r, t ) dt + dw( t ), dv T (t, ) = (b (t) b T (t))(r, t ){b T (t)(r, t) dt + dw (t)}, v T (t, ) (r, t) = 1 (t) + (t)r dw T (t) = b T (t)(r, t) dt + dw (t), dv T (t, ) = (b (t) b T (t))(r, t)dw T (t), 0 t T, v T (t, ) W T (t) Pr T v T (t, )

53 MBS trinomial tree {r (t), t 0} dr (t) = ((t)a(t)r(t)) dt + (r (t), t)dw (t), t 0, a(t)(t)(r (t), t) dx(t) = a(t)x(t)dt + (X(t), t)dw (t), t 0, X (0)= 0, {X(t), t 0} (t) r(t) = (t) + X(t), t 0, (0) = r (0), dr (t) = { (t)+a(t)(t)a(t)r (t)}dt + (r (t), t)dw (t), (t) = (t)+a(t)(t), (r (t), t) = (X(t), t), (t) = exp t ( t ) a( u) du 0 0 t s exp a( u) du ( s) ds + r ( 0) 0, (t)shift function {X(t), t 0} (i, j)t i jx(t) x(i, j) r (t)r (i, j) = x(i, j) + (t i )

54 g(t)= 0 g(t) = 0 E exp t i ds r ( s) S ( t i ) = exp + 0 Y ( t i ) Y t i ( ) = exp (1 ) H ( t i ) Lt i + (1 ) H ( t i ), v (0, t i ) = E exp t i r ( s) ds = exp 0 H ( t i )+ H ( t ) i, v (0, t i ) exp ( t i ) Lt i + H 1 H ( t i ), exp H( t i ) Lt i + 1 ( t H i ) = exp ( r t i + (r(0 ) r )( t i ) Lt i = exp + a r L a 1 a 1 t i t i ( i ( ) + t ) 1 t + r ( 0 ) r + i ( ) ( t ) a i + a 1 (t ), i

55 E exp t i ds r ( s) S ( t i ) = exp Z ( t i ) Z t i ( ) = v ( 0, t ) exp ( t i ) Lt i + i H 1 1 H ( t i 1) cov( H( t i ), H( t i 1 ), exp H ( t i ) H ( t i 1) Lt i cov( H( t i ), H( t i 1 ) = exp ( r t i + (r (0) r 1 )( t i 1 ) Lt i t a i ( t i ) + ( t ) i a a ( t ) t i 1 ( 1 + e i t i 1 ) ( t i 1) + e a ( t i t i 1) ( i t ) = exp r L a 1 t i1 r ( 0 ) r a ( t 1) + ( ) ( t ) a 1 + e i t i + i 1 + a (t a ( t e i t i 1 ) i 1), g(t)= 0

56 JAFFE ABS MBSJAFFE Black, F., and P. Karasinski, Bond and Option Pricing when Short Rates are Lognormal, Finance Analysts Journal, 1991, pp Brigo, D., and F. Mercurio, Interest Rate Models Theory and Practice, Springer, 001. Collin-Dufresne, P., and R. S. Goldstein, Do Credit Spreads Reflect Stationary Leverage Ratios?, The Journal of Finance 56, 001, pp Cox, J. C., J. E. Ingersoll, and S. A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica 53, 1985, pp Duffie, D., First-to-Default Valuation, Working Paper, Dunn, K. B., and J. J. McConnell, Valuation of GNMA Mortgage-Backed Securities, The Journal of Finance 36, 1981, pp Fabozzi, F. J., The Handbook of Mortgage Backed Securities Fifth Edition, McGraw-Hill, 001. Hayre, L., Salomon Smith Barney Guide to Mortgage-Backed and Asset-Backed Securities, John Wiley & Co., 001. Hull, J., and A. White, Pricing Interest Rate Derivative Securities, The Review of Financial Studies 3, 1990, pp Jegadeesh, N., and X. Ju, A Non-Parametric Prepayment Model and Valuation of Mortgage- Backed Securities, The Journal of Fixed Income 10, 000, pp Kariya, T., and M. Kobayashi, Pricing Mortgage-Backed Securities (MBS) -A Model Describing the Burnout Effect-, Asia-Pacific Financial Markets 7, 1999, pp Kijima, M., and Y. Muromachi, Credit Events and the Valuation of Credit Derivatives of Basket Type, Review of Derivatives Research 4, 000, pp Musiela, M., and M. Rutkowski, Martingale Methods in Financial Modelling, Springer, 1997.

57 McConnell, J. J., and M. Singh, Rational Prepayments and the Valuation of Collateralized Mortgage Obligations, The Journal of Finance 49, 1994, pp Nakamura, N., Valuation of Mortgage-Backed Securities Based upon a Structural Approach, Asia-Pacific Financial Markets 8, 001, pp Sandmann, K., and D. Sondermann, A Note on the Stability of Lognormal Interest Rate Models and the Pricing of Eurodollar Futures, Mathematical Finance 7, 1997, pp Schwartz, E. S., and W. N. Torous, Prepayment and the Valuation of Mortgage-Backed Securities, The Journal of Finance 44, 1989, pp Stanton, R., Rational Prepayment and the Valuation of Mortgage-Backed Securities, The Review of Financial Studies 8, 1995, pp Sugimura, T., A Prepayment Model for the Japanese Mortgage Loan Market: Prepayment-Type-Specific Parametric Model Approach, Asia-Pacific Financial Markets 9, 00, pp Vasicek, O. A., An Equilibrium Characterization of the Term Structure, Journal of Financial Economics 5, 1977, pp

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