○松本委員

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らむふじがわ
5 years ago
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1 CIRJE-J CIRJE hp://

2 Credi Risk Modeling Approaches

3 Absrac This aricle originaes from a speech given by he auhor in he seminar organized by he Securiy Analyss Associaion of Japan (SAAJ) on Sepember fifh of 2003 o commemorae he founding of he Cerified Inernaional Invesmen Analys (CIIA) qualificaion. In he firs half, I give a fairly comprehensive, non-quaniaive summary of he recen developmens of credi risk modeling approaches and echniques. In he laer half, I illusrae a new converible-bond (CB) pricing model ha we developed using he reduced-form approach o handle he credi-risk componen embedded in converible bonds. I also presen some resuls of applying our model and, for comparison, a srucural model, o Japanese CB markes. 1

5 CIIA 2 Journal of Fixed Income CIIA 1 1. (1) Srucural Approach 1974 Meron [1974] 1980 KMV Moody s KMV 1 3

6 4 T T Meron [1974] T 1 T T 1 (2) KMV

7 1 T (3) KMV D T 1 T D A T D µ γ γ γ σ A T survival probabiliy (1) pt (, ) 5

8 A T A T D 2 (1) () ( ut ) pt (, ) = (, ) (1) X + m( T ) ut (, ) = T log( A D) X = σ 2 σ m = µ γ σ ut (, ) u σ X disance o defaul 2 A D σ σ A T 6

9 T (4) D defaul boundary D A D T (2)D (2) { τ τ } pt (, ) = Pr > T > X + m( T ) 2mX X ( ) + m T = e T T (2) (Forward Defaul Rae) f(, T) T T + (3) 0 f(, T ) (3) pt (, ) T { τ τ } Pr < T + T (, ) p(, T + ) p T = pt (, ) pt (, ) pt (, ) f(, T) T (3) Duffie-Singleon[2003] 7

10 3 3 A A Zhou [2001] Duffie and Lando [2001] 8

11 (5) 4 A T D D 4 D D 0 D 0 D 0 D (4) ( d 1 ) D ( ) ( ) ( ) ( ) γ T r T = Ae d De d (4) γ ( T ) r( T ) log Ae De 1 d1 = + σ T σ T 2 d = d σ T (5) = A (5) 9

12 (1) pt (, ) (4) ( d 1 ) T (4)d 1 (1) ut (, ) (1) µ (4) r µ (4) µ r 2 10

13 2 (Capial Asse Pricing Model) 11

14 λ λ * 2 λ λ * λ λ * λ Moody s-kmv R&I λ * CDS Credi Defaul Swap λ λ * λ λ * 2 λ * λ λ * λ λ * λ Moody s-kmv λ * λ * 12

15 2. (1) Reduced Form Approach Defaul Inensiy Modeling Approach (2) N () N(0) 0 N () couning process N () 3 1 (independen incremens) 2 (saionary incremens) 3 Pr ( N( ) = 1 ) = λ + o( ), Pr( N( ) > 1) = o( ) (6) λ 1 13

16 Pr( N( ) = 0) = 1 λ + o( ) (7) λ inensiydefaul inensiy (1) (3) 5 5 λ Defaul 1 λ λ Defaul 1 λ λ Defaul 1 λ 2 3 λ N () τ 1 n τ n (3) N () = 1τ 1 τ 1 g() λexp ( λ), for 0 = > (8) negaive exponenial disribuion 14

17 λ λ λe d = e (9) λ 1 ( λe ) d = (10) 0 λ (9) exp( λ) τ 1 1 λ λ λ = λ = exp( 0.04) = λ { λ(): } T pt (, ) pt (, ) = E e T λ() sds (11) (4) { λ(): } Affine Inensiy (12) ( ) dλ() = k θ λ() d+ JdN() (12) Mean-Revering Inensiy 6 λ() 15

18 6 2 CIR Cox, Ingersoll, and Ross λ() ( ) dλ() = k θ λ() d+ σ λ() dw() (13) Cox-Ingersoll-Ross[1985] CIR 7 16

19 7 ( = 200bp, =0.25, (0)= ) θ κ λ θ (5) 3 8 λ() λ 5 8 1, 2, 3 17

20 8 1 e λ 1 Defaul e λ 1 1 e λ 2 Defaul e λ 2 1 e λ 3 Defaul e λ exp( λ ) T i λ() λ *( ) { λ *( ): 0} { r (): 0} T d 0 (, T) d0(, T) = E e [ ( ) *( )] T r s +λ s ds (14) δ (, T) δ (, T) = E e T r( s) ds (15) 18

21 T 1 1 (15) (14)(15) risk-adjused discoun rae R () (14) R () = r () + λ *() (16) d0(, T) = E e T R( s) ds (17) { r (): 0} { λ *( ): 0} (14) d (, T 0 ) T T r( s) ds λ*( s ) ds = E e E e = δ (, T) p*(, T) (18) p*(, T) face value (14)d 0 (, T) 19

22 100 (17) R () L*( ) R () = r () + λ *() L*() (19) 2 2 (6) 9 20

23 9 3. (1) Takahashi-Kobayashi-Nakagawa[2001] 21

24 1 (equiy risk) λ *( ) λ *( )

25 * λ 0 S (16) L*( ) = 1 23

26 { [ ] λ [ ]} ds() = r() d S(), + * S(), S() d + σs() dw () (20) () S [ (),] r () σ d S (20) λ *[ S ( ), ] (2) (20) F 1 a X { } X = max as( ), F (21) (19) R () R () R () S () V[ S(), ] V[ S ] (), as() (22) 24

27 cp() [ (),] max { cp(), as() } V S (23) (23) pp() [ ] V S(), pp() (24) σ 6 LIBOR r () 11 (calibrae) 25

28 Implied inensiy funcion (credi spread sensiiviy) 0.15 Inensiy bps 200 bps 400 bps (credi spread) bps Sock price (3)

29 Credi spread sensiiviy CB price From he upper line, credi spreads are 50 bps 100 bps 150 bps 200 bps 250 bps 300 bps Sock price CB price year 3 year 4 year 5 year Sock price (4) defaul boundary) D 14 27

30 14 Implied defaul boundary (credi spread sensiiviy) sock price credi spread 15(a)15(b) 1, (a) 110 Inensiy m odel CB pri bps 100 bps 200 bps 400 bps sock price 28

31 15(b) 110 Firs-passage model 90 CB price bps 100 bps 200 bps 400 bps sock price CB price Volailiy sensiiviy (HV=0.4969) volailiy Inensiy model Firs-passage model Marke price 29

32 Black F., and M. Scholes [1973], The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, 81, Cox, J.C., J. Ingersoll, and S. Ross [1985], A Theory of he Term Srucure of Ineres Raes, Economerica, 53, Duffie, D., and D. Lando [2001], Term Srucures of Credi Spreads wih Incomplee Accouning Informaion, Economerica, 69, Duffie, D, and K. J. Singleon [2003], Credi Risk: Pricing, Measuremen and Managemen, Princeon Universiy Press. Meron, R. [1974], On he Pricing of Corporae Deb: The Risk Srucure of Ineres Raes, Journal of Finance, 29, Takahashi, A., T. Kobayashi, and N. Nakagawa [2001], Pricing Converible Bonds wih Defaul Risk, Journal of Fixed Income, 11, Zhou, C., [2001], The Term Srucure of Credi Spreads wih Jump Risk, Journal of Banking and Finance, 25,

デフォルト相関係数のインプライド推計（）

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