CIRJE-J-100 2003 11 CIRJE hp://www.e.u-okyo.ac.jp/cirje/research/03research02dp_j.hml
Credi Risk Modeling Approaches 2003 11 17
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
2003 9 5 2
CIIA 2 Journal of Fixed Income CIIA 1 1. (1) Srucural Approach 1974 Meron [1974] 1980 KMV Moody s KMV 1 3
4 T T Meron [1974] T 1 T T 1 (2) KMV
1 T (3) KMV D T 1 T D A T D µ γ γ γ σ A T survival probabiliy (1) pt (, ) 5
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 2 2 0 ut (, ) u σ X disance o defaul 2 A D σ σ A T 6
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) 3 3 6 7 9 Duffie-Singleon[2003] 7
3 3 A A Zhou [2001] Duffie and Lando [2001] 8
(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) 1 2 2 1 γ ( T ) r( T ) log Ae De 1 d1 = + σ T σ T 2 d = d σ T (5) = A (5) 9
(1) pt (, ) (4) ( d 1 ) T (4)d 1 (1) ut (, ) (1) µ (4) r µ (4) µ r 2 10
2 (Capial Asse Pricing Model) 11
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
2. (1) Reduced Form Approach Defaul Inensiy Modeling Approach (2) N () N(0) 0 N () 0 1 2 3 couning process N () 3 1 (independen incremens) 2 (saionary incremens) 3 Pr ( N( ) = 1 ) = λ + o( ), Pr( N( ) > 1) = o( ) (6) 2 0 1 λ 1 13
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
λ λ λe d = e (9) λ 1 ( λe ) d = (10) 0 λ (9) exp( λ) τ 1 1 λ λ λ = 0.04 1 λ = 25 25 1 exp( 0.04) = 0. 0392 3.92 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
6 2 CIR Cox, Ingersoll, and Ross λ() ( ) dλ() = k θ λ() d+ σ λ() dw() (13) Cox-Ingersoll-Ross[1985] CIR 7 16
7 ( = 200bp, =0.25, (0)= ) θ κ λ θ (5) 3 8 λ() λ 5 8 1, 2, 3 17
8 1 e λ 1 Defaul e λ 1 1 e λ 2 Defaul e λ 2 1 e λ 3 Defaul e λ 3 1 2 3 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
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
100 (17) R () L*( ) R () = r () + λ *() L*() (19) 2 2 (6) 9 20
9 3. (1) Takahashi-Kobayashi-Nakagawa[2001] 21
1 (equiy risk) λ *( ) λ *( ) 2 1 10 22
0 0 10 * λ 0 S (16) L*( ) = 1 23
{ [ ] 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
cp() [ (),] max { cp(), as() } V S (23) (23) pp() [ ] V S(), pp() (24) σ 6 LIBOR r () 11 (calibrae) 25
0.25 0.2 11 Implied inensiy funcion (credi spread sensiiviy) 0.15 Inensiy 0.1 100 bps 200 bps 400 bps (credi spread) 0.05 50 bps 0 0 500 1000 1500 2000 2500 3000 3500 Sock price (3) 12 100 13 26
12 120 Credi spread sensiiviy 115 110 105 CB price 100 95 90 85 80 From he upper line, credi spreads are 50 bps 100 bps 150 bps 200 bps 250 bps 300 bps 75 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Sock price 13 200 180 160 140 CB price 120 100 80 60 40 1year 3 year 4 year 5 year 20 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Sock price (4) defaul boundary) D 14 27
14 Implied defaul boundary (credi spread sensiiviy) 500 400 sock price 300 200 100 0 10 50 100 200 400 credi spread 15(a)15(b) 1,940 98.2 15(a) 110 Inensiy m odel 105 100 CB pri 95 90 85 50 bps 100 bps 200 bps 400 bps 80 500 1000 2000 3000 5000 sock price 28
15(b) 110 Firs-passage model 90 CB price 70 50 30 50 bps 100 bps 200 bps 400 bps 10 500 1000 2000 3000 5000 sock price 16 3 16 CB price Volailiy sensiiviy (HV=0.4969) 160 150 140 130 120 110 100 0.2 0.3 0.4 0.4969 0.6 0.7 0.8 0.9 volailiy Inensiy model Firs-passage model Marke price 29
Black F., and M. Scholes [1973], The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, 81, 637-654. Cox, J.C., J. Ingersoll, and S. Ross [1985], A Theory of he Term Srucure of Ineres Raes, Economerica, 53, 385-407. Duffie, D., and D. Lando [2001], Term Srucures of Credi Spreads wih Incomplee Accouning Informaion, Economerica, 69, 633-664. 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, 449-470. Takahashi, A., T. Kobayashi, and N. Nakagawa [2001], Pricing Converible Bonds wih Defaul Risk, Journal of Fixed Income, 11, 20-29. Zhou, C., [2001], The Term Srucure of Credi Spreads wih Jump Risk, Journal of Banking and Finance, 25, 2015-2040. 30