: Ω F F 0 t T P F 0 t T F 0 P Q. Merton 1974 XT T X T XT. T t. V t t X d T = XT [V t/t ]. τ 0 < τ < X d T = XT I {V τ T } δt XT I {V τ<t } I A

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Download ": Ω F F 0 t T P F 0 t T F 0 P Q. Merton 1974 XT T X T XT. T t. V t t X d T = XT [V t/t ]. τ 0 < τ < X d T = XT I {V τ T } δt XT I {V τ<t } I A"

Λευί Μαυρίδης
5 χρόνια πριν
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1 Chinese Journal of Applied Probability and Statistics Vol.28 No.2 Apr :. : O Johnson Stulz [3] Merton 1974 Johnson Stulz Hull White 1995 Klein Klein Klein

2 : Ω F F 0 t T P F 0 t T F 0 P Q. Merton 1974 XT T X T XT. T t. V t t X d T = XT [V t/t ]. τ 0 < τ < X d T = XT I {V τ T } δt XT I {V τ<t } I A A δt δt = V T /T. Q t < T X d T = B t E Q [B 1 T XT I {V τ T } δt XT I {V τ<t } F t ] B t. t r Q St V t t dst St = rdt k Sd npn S t σ S dw S t J S dnt S 2.1 dv t V t = rdt k V d npn V t σ V dw V t J V dnt V 2.2 r ; σ 2 S σ2 V ; W S t W V t Q Brown ρ; Nt S Nt V λ S λ V Possion ; W S t W V t Nt S Nt V ; k S d npn S t k V d npn V t Possion. K S = EJ S K V = EJ V J S J V Possion J S > 1 J V > 1 J S 0 = 0 J V 0 = 0. oleans-ade [6] N t S St = S0 1 Ji S exp r 1 2 σ2 S t k S npn S t σ S W S t

3 174 Nt V V t = V 0 1 Ji V exp r 1 2 σ2 V t k V npn V t σ V W V t. Q t < T B-S XT = ST K Xt = StNd KNd σ exp r X t = B t E Q [B 1 T ST K I {V τ T } δt XT I {V τ<t } F t ] Y T = K ST Y t = B t E Q [B 1 T K ST I {V τ T } δt XT I {V τ<t } F t ]. r B t = exprt t = : 3.1 XT = ST K δt XT X t X t = P mn tε mn {StN 2 a 1 a 2 ρ m 1Ji S exp k S ipi S StV t KN 2 b 1 b 2 ρ exp rt t N 2c 1 c 2 ρ m 1J S i n 1Ji V exp r ρσ S σ V k S ipi S k V ipi V KV t N 2d 1 d 2 ρ n } 1 Ji V exp k V ipi V ε mn m 1 Ji S n 1 Ji V P mn t = expλ S λ V λm S λn V mn m!n! a 1 = e S1 e S3 a 2 = e V1 e V2 e V3 b 1 = e S1 e S3 b 2 = e V1 e V2 e V3 σ S c 1 = e S1 e S2 e S3 c 2 = e V1 e V2 e V3 d 1 = e S1 e S2 e S3 d 2 = e V1 e V3

4 : 175 e S1 = e V1 = ln St K ln V t m ln1 Ji S r k S ipi S σ S m ln1 Ji V r k V ipi V σ V e S2 = ρσ V ev2 = ρσ S es3 = 1 2 σ S ev3 = 1 2 σ V d SV N 2 S V N 2 x 1 x 2 = d SV z 1 z 2 ρ = : x1 x2 d SV z 1 z 2 ρdz 1 dz 2 1 2πσ S σ exp 1 [ z 2 1 V 1 ρ 2 21 ρ 2 σs 2 z2 2 σ 2 V 2ρz 1z 2 σ S σ V ]. X T = B t E Q [B 1 T ST K I {V T T } δt XT I {V T <T } F t ]. CST V T T = ST K I {V T } δt XT I {V T <} = fst V T CST V T t = exp re Q [fst V T ] = exp r P mn t N 2 ρdf = exp r P mn t N 2 ρdf F. Xi T = E 1 E 2 E 3 E 4 E 1 = B t E Q [B 1 T ST I {ST >K}I {V T } F t ] E 2 = B t E Q [B 1 T KI {ST >K}I {V T } F t ] E 3 = B t E Q [B 1 T ST δt I {ST >K}I {V T <} F t ] E 4 = B t E Q [B 1 T E 1 : KδT I {ST >K}I {V T <} F t ] B t E Q [B 1 T F t] = exp r E 1 = exp re Q [ST I {ST >K} I {V T } F t ] = { P mn tε mn St m 1 Ji S I {ST >K} I {V T } exp k S = ip S i { P mn tε mn exp k S ip S i fz 1 z 2 ρdz 1 dz 2 } St m 1 Ji V I {ST >K} I {V T } } fv 1 v 2 ρdv 1 dv 2

5 176 v 1 = z 1 σ S v2 = z 2 ρσ S N 2 σ S ρσs ρ. L 1 = I {ST >K} I {V T >} fv 1 v 2 ρdv 1 dv 2 d Q dq = exp γ W T 1 2 σ2 T γ W R 2 λ S = σ S λ V = σ V. Girsanov W t = W t γt Q R 2. z = W T W t = W T W t γ = z γ E 1 Z 1 Z 2 Q. E 1 E 1 = StN 2 a 1 a 2 ρ N 2 a 1 a 2. E Q [I {ST >K} ] = QST > K = Qln ST > ln K = Q { σ S W T W t > ln St N S K T = Q r 1 2 σ2 S k S ipi S { z 1 < ln St N S K T i=n S t 1 ln1 J S i = Q{ z 1 < e S1 e S3 } = Q{ z 1 < a 1 }; ln1 Ji S i=nt S } 1 r 1 2 σ2 S k S σ S E Q [I {V T >} ] = QV T > = Qln V T > ln = Q { σ V W T W t > ln V t N V K T ln1 Ji V i=nt V 1 r 1 2 σ2 V ρσ S σ V k V ip V i } ip S i } { ln V t N V = Q K T ln1 Ji V r 1 i=nt z 1 < V 1 2 σ2 V k V ipi V } σ V = Q{ z 2 < e V1 e V2 e V3 } = Q{ z 2 < a 2 }.

6 : 177 [2] = = a 1 a1 a2 E 1 = E 2 E 3 E 4. I {ST >a1 }I {V T >a2 }fv 1 v 2 ρdv 1 dv 2 a 2 fz 1 z 2 ρdz 1 dz 2 = fz 1 z 2 ρdz 1 dz 2. [ P mn tε mn St m a2 a 1 1 Ji S exp k S fz 1 z 2 ρdz 1 dz 2 ] ipi S Na 1 a 2 ρ. 3.2 Y T = K ST δt Y T Y t Y t = P mn tε mn { St m 1 Ji S N 2 a 1 a 2 ρ exp k S KN 2 b 1 b 2 ρ exp r KV t N 2 c 1 c 2 ρ m 1 J S i n 1 Ji V exp r ρσ S σ V k S ipi S k V KV t 3.1. N 2 d 1 d 2 ρ n 1 Ji V exp k V : ipi S ip V i ipi V } Q dt t = rdt σ dw t W T Q Brown. t = t exp r 1 2 σ2 σ W T W t.

7 178 S V A σs 2 ρ SV σ S σ V ρ S σ S σ A = ρ SV σ S σ V σv 2 ρ V σ V σ ρ S σ S σ ρ V σ V σ σ 2 ρ SV S V. δt = V t/t dδ δ = dv V dv /V dv /V d d 2 dv V d dδ δ = σ V dw V t σ dw t ρ V σ V σ dt k V d ip V i t J V dn V t. σ δ W δ t = σ V W V t σ W t σ 2 δ = σ2 V σ2 2ρ V σ V σ. ean-lease δ T = δ t N V T 1 J V i=nt v1 i exp σ ρ 2 V σ V σ 1 2 σ2 δ T t k V d = δ t N V T : 1 J V i=nt v1 i exp 1 2 σ2 1 2 σ2 δ k V d ipi V tσ δ W δ T W δ t ipi V t σ δ W δ T W δ t 3.3 Y T = K ST δt XT X T X t = P mn tε mn {StN 2 a 1 a 2 ρ m 1Ji S exp k S ipi S T t KN 2 b 1 b StV t 2 ρ exp rt t N 2c 1 c 2 ρ m 1J S i n 1Ji V exp ψs V k S ipi S k V ipi V KV t N 2d 1 d 2 ρ n } 1 Ji V exp ϕv k V ipi V ψs V = ρ SV σ S σ V ρ S σ S σ ρ V σ V σ σ 2 ϕv = r σ2 ρ V σ V σ a 1 = e S1 2r σ2 S a 2 = e δ1 2σ σ2 V σ2 2θ S σ δ

8 : 179 b 1 = e S1 2r σ2 S b 2 = e δ1 2σ σ2 V σ2 S 2σ δ c 1 = e S1 2r σ2 S 2θ c 2 = e δ1 2σ σ2 V 3σ2 2θ 4ρ V σ V σ S 2σ δ d 1 = e S1 2r σ2 S 2θ d 2 = e δ1 2σ σ2 V 3σ2 4ρ V σ V σ S 2σ δ ln St e K m ln1 Ji S k S ipi S S1 = σ S V t ln e δ1 = m ln1 Ji V k V ipi V σ δ ρ = ρ SV σ V ρ S σ σ δ σ 2 δ = σ2 V σ 2 2ρ V C σ V σ θ = ρ SV σ S σ V ρ S σ S σ. : Y T = K ST δt Y T Y t Y t = P mn tε mn { K m 1 Ji S N 2 a 1 a 2 ρ exp k S KN 2 b 1 b 2 ρ exp r KV t N 2 c 1 c 2 ρ m 1 J S i n 1 Ji V exp ψs V k S ipi S k V KV t 3.3. ipi S N 2 d 1 d 2 ρ n 1 Ji V exp ϕv k V : ipi V } ipi V. : 1

9 180 J V = ; 2 J S = 0 J = 0 Klein 1996 ; 3 J S = 0 J S = 0 σ 0 Merton 1974 ; 4 V t E 3 E 4 N 2 x 1 ρ = 0 N 2 x 1 ρ = Nx 1 a 2 b 2 c 2 d 2 B-S. [1] Merton R. On the pricing of corporate debt: The risk structure of interest rates Journal of Finance [2] Abramowitz M. and Stegun. I.A. Handbook of Mathematical Functions over Publications [3] Johnson H. and Stulz R. The pricing of options under default risk Journal of Finance [4] Hull J.M. and White A. The impact of default risk on default risk on the prices of options and other derivative securities Journal of Banking and Finance [5] Klein P. Pricing Black-Scholes options with correlated credit risk Journal of Banking and Finance [6] [7] Vulnerable European Option Pricing for Two Jump -iffusion Processes Yan ingqi Yan Bo School of Mathematics and Statistics Lanzhou University Lanzhou The pricing of the derivatives associated with counterparty default risk is considered. Based on Merton s structured credit risk model an explicit pricing formula of vulnerable options was derived when the underlying asset price and corporate value is assumed to follow a jump-diffusion process. A model of vulnerable option pricing is developed when the underlying asset price and corporate value is assumed to follow a jump-diffusion process then the pricing of vulnerable option is discussed when the corporate liabilities are fixed and random were derived respectively. Keywords: Credit risks vulnerable European option pricing corporate value jump-diffusion process. AMS Subject Classification: 60K99.

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