Vulnerable European option pricing with the time-dependent for double jump-diffusion process

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Vulnerable European option pricing with the time-dependent for double jump-diffusion process"

Ἀριστόδημε Κυπραίος
5 χρόνια πριν
Προβολές:

1 Journal of Eas China Normal Universiy Naural Science No. 1 Jan. 014 : ,, 1116 :,., r σ d ;, Iô,. : ; ; : O11.6 : A DOI: /j.issn Vulnerable European opion pricing wih he ime-dependen for double jump-diffusion process LYU Li-juan, ZHANG Xing-yong College of Sciences, China Universiy of Mining and Technology, Xuzhou Jiangsu 1116, China Absrac: Based on Meron s srucured credi risk model, derivaives pricing wih rival unilaeral defaul risk was sudied in his paper. Assuming ha underlying asse price and asses-liabiliies of sellers follow double jump-diffusion process, where risk-free ineres rae r, volailiy of asse σ and dividend yield d are ime-dependen, vulnerable European opions pricing model under double jump-diffusion process was esablished using he srucured mehod, he analyical expressions of opions price was obained using Iô lemma and he runformaion of he equivalen maringale measure. Key words: wo jump-diffusion process; credi risks; vulnerable European opion pricing 0,.,..,,.., : : JGK101658; 013DXS0 :,,,. puyang085@16.com.

2 , : [1-5],. [6-8],,, ;,. Johnoson Sulz [9] Meron,. Hull Whie [10],. Klein [11],. Lobo [1] Zhou [13],. Liu [14] Klein. Klein Yang [15] Johnoson- Sulz Klein., [16] -,. [17], -. Xu [18]. [19,0] C-N,.,,. -; -,,. 1 W = W S, W δ Ω, F, F 0T, P Brown, ρ, dw S, dw δ = ρ d.,, Q, S, δ, -, ds = r d λβ S d + σ S dw S + k dq, S 1 dδ = r θβ δ d + σ δ dw δ + l dp, δ r, d σ S, σ δ [0, + R,. q, p λ, θ Possion, dw S, dq, dw δ, dp, k, l, 1 + k 1 + l, P1 + k 0 = 0, ln1 + k N α σ S, σ S, 3 P1 + l 0 = 0, ln1 + l N γ σ δ, σ δ. 4

3 1, : - 15 E[expln1 + k 1] = expα 1 = β S, 5 E[expln1 + l 1] = expγ 1 = β δ, 6 Q, Iô dlns = r d λβ S σ S d + σ S dw S + ln1 + kdq, dlnδ = r θβ δ σ δ d + σ δ dw δ + ln1 + ldp., Q Doléase-Dade S T = S e δ T = δ e ru du λβ S σ S u du+ qt q σsu dwsu+ P ru θβ δ σ δ u du+ pt p T σ P δu dw δ u+ j=0 µ S : = E Q [lns T /S F, qt q = m] = ru du λβ S σ S u du + m α 1 ν S : = Var Q [lns T /S F, qt q = m] = σ S udu + m µ δ : = E Q [lnδ T /δ F, pt p = n] T = ru θβδ σ δ u du + n γ 1 ν δ : = Var Q [lnδ T /δ F, pt p = n] = σ δ udu + n i=0 ln1+k i, 7 ln1+l j. 8 σ Sudu, 9 σs udu, 10 σ δudu, 11 σδ udu. 1, Q, lns T /S,lnδ T /δ, ln S T /S Nµ S, ν S, ln δ T /δ Nµ δ, ν δ. 13 Q, S, K T, 0 T, V T T V T = S T K +,, V d T = V T I {δt 1} + δ T V T I {δt <1}, 14

4 I A A. B 1 T S T B 1 T δ T Ω, F, F 0T, Q,, V = B E Q [B 1 T S T K + I {δt 1} + δ T I {δt <1}]. 15 Y = B E Q [B 1 T K S T + I {δt 1} + δ T I {δt <1}]. 16.1, V T = S T K + V = m=0 n=0 [λt ] m [θt ] n e λ+θt E 11 E 1 + E 13 E m! n! E 11 = e rudu S e µs+ ν S N a 1, a, ρ, E 1 = e rudu KN b 1, b, ρ, E 13 = e rudu S δ e µs+µ δ+ ν S +ν δ +ρ ν S+ν δ N c 1, c, ρ, E 14 = e rudu Kδ e µ δ+ ν δ N d 1, d, ρ. a 1 = 1 ln S K + µ S + ν S b 1 = 1 ln S K + µ S c 1 = 1 ln S d 1 = 1 ln S, a = 1 νδ ln δ + µ δ + ρ ν S ν δ,, b = 1 K + µ S + ν S + ρ ν S ν δ K + µ S + ρ ν S ν δ 1 N a, b, ρ = π 1 ρ fx, y, ρ = a b νδ ln δ + µ δ, 1 π 1 ρ e 1 1 ρ x ρxy+y., c = 1 lnδ + µ δ + ν δ + ρ ν S ν δ, νδ, d = 1 lnδ + µ δ + ν δ, νδ e 1 1 ρ x ρxy+y dxdy, µ S, ν S, µ δ, ν δ 9 1. Q, B, V = B E Q [ B 1 T S T K + I {δt 1} + δ T I {δt <1} ], 18 db = rb d, B T = 1. 19

5 1, : - 17, V S, δ, = e rudu E Q [S T K + I {δt 1} + δ T I {δt <1}] = e rudu [λt ] m [θt ] n e λ+θt N +, +, ρdf, 0 m! n! F. m=0 n=0 E 11 = B E Q [B 1 T S TI {ST >K}I {δt 1} F ], E 13 = B E Q [B 1 T S Tδ T I {ST >K}I {δt <1} F ], E 1 = B E Q [B 1 T KI {S T >K}I {δt 1} F ], E 14 = B E Q [B 1 T Kδ TI {ST >K}I {δt <1} F ]. V = E 11 m=0 n=0 7, 9 10, [λt ] m [θt ] n e λ+θt E 11 E 1 + E 13 E 14, 1 m! n! E 11 = e rudu E Q [S T I {ST >K}I {δt 1} F ], E 11 = e ru du + z 1 = 1 + ru du λβ S σ S u S e µs+ ν Sz 1 I {ST >K}I {δt 1}fz 1, z, ρdz 1 dz. 3 du qt q + σ S udw S u + ln1 + k i µ S, z = 1 i=0 ru θβ δ σ δ u pt p du + σ δ udw δ u + j=0 ln1 + l j µ δ. z 1, z, ρ. u 1 = z 1 ν S, u = z ρ ν S, E 11 = e rudu + Q, Θ = Θ1 Θ + S e µs+ υ S I{ST >K}I {δt 1}fu 1, u, ρdz 1 dz. 4 d Q dq = T z eθ 1 Θ T, 5 = υs T ρ υs T, z = z1 z, z = z1 z. 6

6 Girsanov, z = z Θ T Q. Q E e [ I {ST >K}] = Q[ST > K] = [S Q e µs+ υs ] υ S ez 1+ρq T T > K [ = Q z < 1 ln S ] υs K + µ S + υ S = Q[ z < a 1 ]. 7 E eq [I δt 1] = Q [δ T 1] = [δ Q e µ δ+ υδ υ δ ez +ρq T T ] = Q [ z 1 υδ lnδ + µ δ + ρ υ S υ δ ] 1 = Q [ z a ]. 8, = a I {z1 a 1}I {z a }fz 1, z, ρdz dz 1 = fz 1, z, ρdz dz 1 = a1 a a a 1 a fz 1, z, ρdz dz 1 fz 1, z, ρdz dz 1, 9 E 11 = e rudu S e us+ υ S N a 1, a, ρ. 30,,.., Y T = K S T + Y = m=0 n=0 [λt ] m [θt ] n e λ+θt E 1 E + E 3 E m! n! E 1 = e rudu S e us+ υ S N a 1, a, ρ, E = e rudu KN b 1, b, ρ, E 3 = e rudu S δ e µs+µ δ+ υ S +υ δ +ρ υ S+υ δ N c 1, c, ρ, E 4 = e rudu Kδ e µ δ+ υ δ N d 1, d, ρ..1. 3, λ =, θ = 0.5, r = 0.07, d = 0.0, σ S = 0.6, σ δ = 0.3, α = 0.18, γ = 0.045, K = 40, ρ = 0.5, T = 0.8, = 0.

7 1, : ,,, -,., - θ = 0,, -, θ Fig. 1 Influence of parameer θ on opion pricing ρ Fig. Influence of parameer ρ on opion pricing, Brown ρ.,, ρ.,.,. -. 3, -,, -,, 1.

0 014 - Fig. Influence of asse price and asses-liabiliies on opion pricing 4,, -, Zhou Lobo,,.,,. [ ] [ 1 ] BLACK F, SCHOLES M. The valuaion of opions and corporae liabiliies[j].

8 Fig. Influence of asse price and asses-liabiliies on opion pricing 4,, -, Zhou Lobo,,.,,. [ ] [ 1 ] BLACK F, SCHOLES M. The valuaion of opions and corporae liabiliies[j]. Journal of Poliical Economy, 1973, 8: [ ] MERTON R C. On he pricing of corporae deb:he risk srucure of ineres raes[j]. Journal of Finance, 1974, 9: [ 3 ] BLACK F, COX J. Valuing corporae securiies: some effecs of bond indenure provisions[j]. Journal of Finance, 1976, 11: [ 4 ] LONGSTAFF F, SCHWARTZ E. A simple approach o valuing risky fixed and floaing rae deb[j]. Journal of Finance, 1995, 50: [ 5 ] YILDIRIM Y. Modeling defaul risk: a new srucural approach[j]. Finance Research Leers, 006, 3: [ 6 ] DUFFIE D, SINGLETON K. Modelling erm srucures of defaulable bonds[j]. Review of Financial Sudies, 1996, 1: [ 7 ] JARROW R, LANDO D, TURNBULL S. A Markov model for he erm srucure of credi spreads[j]. Review of Financial Sudies, 19971: [ 8 ] MADAN D B, UNAL H. Pricing he risks of defaul[j]. Review of Derivaives Research, 1998: [ 9 ] JONHNSON H, STULZ R. The pricing of opions under defaul risk[j]. Journal of Finance, 1987, 4: [10] HULL J M, WHITE A. The impac of defaul risk on defaul risk on he prices of opions and oher dervaive securiies[j]. Journal of Banking and Finance, 1995, 19: [11] KLEIN P. Pricing Black-Sholes poions wih correlaed credi risk[j]. Journal of Banking and Finance, 1996, 0: [1] LOBO B J. Jump risk in he U.S.sock marke: evidence using poliical informaion[j]. Review of Financial Economics, 19998: [13] ZHOU C. The erm srucure of credi spread wih jump risk[j]. Journal of Banking and Finance, 001, 5:

9 6 014 [ 5 ],. [J]. :, 00, 61: 1-7. [ 6 ],. [J], 006, 7A4: [ 7 ],. [J], 009, 9A4: [ 8 ], [J] :, 006, 455: [ 9 ],,. [M]. :, [10] CHEN Z X. Zeros of meromorphic soluions of higher order linear differenial equaions[j] Analysis, 1994,14: [11]. f + e z f + QZf = 0 [J]. : A, 001, 319: [14] LIU S I, LIU Y C. Pricing Vulnerable Opions by Binomial Trees[R]//Working Paper. Taipei: Shih Hsin Universiy, 006. [15] KLEIN P, YANG J. Vulnerable American opions[j]. Managerial Finance, 010, 36: [16]. - [J]., 0081: [17],. -[J]., 01: [18] XU W D, XU W J, LI H Y, e al. A jump-diffusion approach o modelling vulnerable opion pricing[j]. Finance Research Leers, 019: [19],. [J]. :, 014: [0],,,. [J]. :, 0115: 5-3.

Σχετικά έγγραφα

: 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

: 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 2012 4 Chinese Journal of Applied Probability and Statistics Vol.28 No.2 Apr. 2012 730000. :. : O211.9. 1..... Johnson Stulz [3] 1987. Merton 1974 Johnson Stulz 1987. Hull White 1995 Klein 1996 2008 Klein