4 3 2 Vol 43 No 2 2 1 4 4 Journal of Shanghai Normal UniversityNatural Sciences Apr 2 1 4 DOI1 3969 /J 1SSN 1-5137 214 2 2 1 2 2 1 22342 2234 O 175 2 A 1-51372142-117-1 2 7 8 1 2 3 Black-Scholes-Merton Ingersoll1977 1 Brennan Schwartz1977 2 Brennan Schwartz198 3 Barone 23 4 Toshikazu26 5 Monte Carlo 214-2-28 13ZZ17 E-mailfuyi@ shedu edu cn
118 214 Kostas 1998 6 Ayache23 7 Wang21 8 Kyoko21 9 ahuai211 1 Toshikazu 26 5 21 11 29 12 213 13 1 1 A T B 2 A B 3 λ 1 λ 2 1 2 4 Poisson Poisson s t k PN t - N s = k= 1 k λk t - s k e -λt -s T P T= 1 - P > T= 1 - e -λt 5 6 St dst= μstdt + σstdwt μ σ Wt 7 S + = S - 1 - η η 8
2 119 9 k q 2 Δ - Π t = V t - ΔS t V S t t + dt dπ = dv - ΔdS = V V + μs t S + 1 2 V ( 2 σ2 S 2 + q) dt + σs V 2 S S dw t - ΔμSdt + σsdw t Δ = V S dπ = V t + 1 2 V ( 2 σ2 S 2 + q) dt 2 S t t + dt t t + dt λ 1 dt 7 1 R R 1 R + 1 - R 2 ks1 - η 1 η 1 Π dπ = 1 - λ 1 dt V t + 1 2 V ( 2 σ2 S 2 + q) dt + λ S 2 1 dtδsη 1 - V+ λ 1 dtmax ks1 - η 1 1 dπ = rπdt 2 1 2Ito^ 1 - λ 1 dt V t + 1 2 V ( 2 σ2 S 2 + q) dt + λ S 2 1 dtδsη 1 - V+ λ 1 dtmax ks1 - η 1 = rπdt V t + r + λ 1η 1 S V S + 1 2 V 2 σ2 S 2 S - λ 2 1 + rv + λ 1 max ks1 - η 1 + q = V t + r + λ1η S V 1 S + 1 2 V 2 σ2 S 2 S - λ 2 1 + rv + λ 1 max ks1 - η 1 + q = 3 V t = T = maxks t t + dt λ 1 dt λ 2 dt 1 R R
12 214 2 ks1 - η 2 η 2 > η 1 η 2 Π ( ) dπ = 1 - λ 1 dt V t + 1 2 V 2 σ2 S 2 + q dt + λ S 2 1 dtδsη 2 - V+ λ 1 dtmaxr ks1 - η 2 1 - λ 1 dt V t + 1 2 V ( 2 σ2 S 2 + q) dt + λ S 2 1 dtδsη 2 - V+ λ 1 dtmaxr ks1 - η 2 = r( V - S V ) S dt V t + r + λ1η S V 2 S + 1 2 V 2 σ2 S 2 S - r + λ 1V + λ 2 1 maxr ks1 - η 2 + q = 4 V t = T = max ks 2 1 < 2 T 1 2 T 4 1 2 3 2 2 > T T 3 1 < 2 T 1 2 P 1 T = P 2 T 1 2 = T 1 λ 1 f 1 2 d 2 d 1 = 1 + e -λ1 +λ2t-1 - e -λ1t 5 λ 1 + λ 2 2 < 2 T 1 2 T P 2 T = P 2 T 1 2 = 2 λ 2 f 1 2 d 2 d 1 + T T λ 1 λ 2 e -λ 1 1 -λ 2 2 d 2 d 1 = λ 1 + λ 2 1 - e -λ1 +λ2t 6 3 2 > T P 3 = P 2 > T= e -λ 2 T 7 3 34 3 x = S = T - t 3 Cauchy V - ( r + λ 1η 1 - ) σ2 V 2 x - 1 2 V 2 σ2 x + λ 2 1 + rv - λ 1 max ke x 1 - η 1 - q = V = = maxke x V = Ue α + βx α β U - 1 2 U 2 σ2 x - e -α -βx λ 2 1 max ke x 1 - η 1 + q= 8 Ux = e -βx maxke x
2 121 ( 1 ) α = - λ 1 - r - 1 1 2σ 2 2 σ2 - r - λ 1 η 2β = 1 2 - r σ - λ 1η 1 2 σ 2 8 U Ux = e -βx maxke x 9 9 1 Ux = σ 槡 2π - e -x-ξ 2σ2 -βξ maxke ξ dξ 1 U - 1 2 U 2 σ2 - e -α -βx λ x 2 1 max ke x 1 - η 1 + q= Ux = 11 11 1 Ux = σ 槡 2π λ 1 max ke ξ 1 - η 1 + q e - x-ξ2 2σ2 -ζ -αζ-βξ dξdζ - 槡 - ζ 12 112 8 1 Ux = e -x-ξ σ 槡 2π 2σ2 -βξ maxke ξ dξ + - 1 λ 1 max ke ξ 1 - η 1 + q σ 2 槡 π - - 槡 Ux = e σ2β-x2 -x2 2σ2 Nd 1 + ke σ 2β-x-σ22 -x2 2σ2 1 - Nd 2 + λ 1 + q q ζ e -x2 +2ασ2 -ζζ -βσ2 -ζ-x2 2σ2 -ζ Nd 3 dζ + e -x2 +2ασ2 -ζζ -βσ2 -ζ-x2 2σ2 -ζ 1 - Nd 3 dζ + λ 1 k1 - η 1 e - x-ξ2 2σ2 -ζ -αζ-βξ dξdζ e -x2 +2ασ2 -ζζ -β-1σ2 -ζ-x2 2σ2 -ζ 1 - Nd 4 dζ d 1 = d 2 = d 3 = k + σ2 β - x σ 槡 k + σ2 β - x - σ 2 σ 槡 k1 - η 1 + βσ2 - ζ- x σ 槡 - ζ
122 214 k1 - η 1 + β - 1σ2 - ζ- x d 4 = 13 σ 槡 - ζ VS t V 1 S V 1 S t= e α+βs e σ2β-s2 -S2 2σ2 Nd' 1 + ke σ 2β-S-σ22 -S2 2σ2 1 - Nd' 2 + λ 1 + q q e -S2 +2ασ2 -ζζ -βσ2-ζ-s2 2σ2 -ζ Nd' 3 dζ + e -S2 +2ασ2 -ζζ -βσ2 -ζ-s2 2σ2-ζ 1 - Nd' 3 dζ + λ 1 k1 - η 1 e -S2 +2ασ2-ζζ -β-1σ2 -ζ-s2 2σ2 -ζ 1 - Nd' 4 dζ d' 1 = k + σ2 βt - t- S σ 槡 T - t 4 d' 2 = d' 3 = d' 4 = k + σ2 βt - t- S - σ 2 T - t σ 槡 T - t k1 - η 1 + σ2 βt - t - ζ- S σ 槡 T - t - ζ k1 - η 1 + σ2 β - 1 T - t - ζ- S σ T - t - 槡 Wx = e σ2β-x2 -x2 2σ2 Nd 1 + ke σ 2β -x-σ22 -x2 2σ2 1 - Nd 2 + λ 1 R + q q ζ e -x2 +2ασ2 -ζζ -βσ2 -ζ-x2 2σ2 -ζ Nd 5 dζ + e -x2 +2ασ2 -ζζ -βσ2 -ζ-x2 2σ2 -ζ 1 - Nd 5 dζ + λ 1 k1 - η 2 e -x2 +2ασ2 -ζζ -β-1σ2 -ζ-x2 2σ2 -ζ 1 - Nd 6 dζ 14 d 1 d 2 13 k1 - η 2 + βσ2 - ζ- x d 5 = σ 槡 - ζ k1 - η 2 + β - 1σ2 - ζ- x d 6 = σ 槡 - ζ VS t V 2 S V 2 S t= e α+βs e σ2β-s2 -S2 2σ2 Nd' 1 + ke σ 2β-S-σ22 -S2 2σ2 1 - Nd' 2 +
2 123 λ 1 R + q q e -S2 +2ασ2 -ζζ -βσ2 -ζ-s2 2σ2 -ζ Nd' 5 dζ + e -S2 +2ασ2 -ζζ -βσ2 -ζ-s2 2σ2 -ζ 1 - Nd' 5 dζ + λ 1 k1 - η 2 d' 1 d' 2 14 e -S2 +2ασ2-ζζ -β -1σ2 -ζ-s2 2σ2 -ζ 1 - Nd' 6 dζ d' 5 k1 - η 2 + βσ2 T - t - ζ- S = σ 槡 T - t - ζ d' 6 k1 - η 2 + β - 1σ2 T - t - ζ- S = σ 槡 T - t - ζ VS t 4 VS t= P 1 + P 3 V 1 S t+ P 2 V 2 S t 1 1 V t S 2 λ 1 2 λ 1 3 4 λ 2 3 λ 1 = 5 λ 2 λ 2 4 λ 1 = 3 λ 2 λ 2 5 r 5 r λ 1
124 214 2 λ 1 3 λ 1 = 5 λ 2 4 λ 1 = 3 λ 2 5 r 6 σ 6 σ 6 σ 7 R
2 125 7 R 7 8 q 8 9 k 9 8 q 9 k 5 AV 1INGERSOLL JR J A contingent-claims valuation of convertible securities J Journal of inancal Economics 1977 4289-322 2BRENNAN M J SCHWARTZ E S Convertible bondsvaluation and optimal strategies for call and conversion J Journal of inance 1977 321699-1715 3BRENNAN M J SCHWART E S Analyzing convertible bondsj Journal of inance and Quantitative Analysis 198 1597-929 4BARONE ADESI G BERMUDEZ A HATGIOANNIDES J Two factor convertible bonds valuation using the method of characteristics finite elements J Journal of Economic Dynamics and Control 23 27181-1831 5KIMURA T SHINOHARA T Monte carlo analysis of convertible bonds with reset clauses J European Journal of Operational Research 26 16831-31 6TSIVERIOTIS K ERNANDES C Valuing convertible bonds with credit riskj Journal of ixed Income 1998 895-12 7AYACHE E ORSYTH P A VETZAL K R The valuation of convertible bonds with credit risk J The Journal of Derivatives 23 119-3 8WANG L L BIAN B J Pricing of perpetual convertible bonds with credit risk under framework of reduce form J Journal
126 214 of Tongji University 21 6935-94 9KYOKO Y KATSUSHIGE S The valuation of callable-puttable reverse convertible bonds J Asia-Pacific Journal of Operational Research 21 27189-29 1ZHOU Y YI H A free boundary problem arising from pricing convertible bond J Applicable Analysis 21 337-323 11 J 21 233-39 12 J 29 7989-992 13 J 213 3465-469 The research of convertible bond pricing based on credit risk of guarantor U Yi 1 ZHANG Jizhou 2 QIU Yazun 2 1 Business School Shanghai Normal University Shanghai 2234 China 2 College of Mathemcties and Sciences Shanghai Normal University Shanghai 2234 China AbstractIn this paper we study convertible bonds pricing from the perspective of investors under the background of credit risk of guarantor The default process of the bond issuer and guarantor are assumed to be a poisson process and we consider the stock price jumps after the issuer default Through the hedge we get the party differential equation model and the explicit solution inally we calculate the solution and analysis the effect of various parameters in the model Key wordsconvertible bondsguaranteereduced form methodcredit risk