DOI /J. 1SSN

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1 4 3 2 Vol 43 No Journal of Shanghai Normal UniversityNatural Sciences Apr DOI /J 1SSN O A Black-Scholes-Merton Ingersoll Brennan Schwartz Brennan Schwartz198 3 Barone 23 4 Toshikazu26 5 Monte Carlo ZZ17 fuyi@ shedu edu cn

2 Kostas Ayache23 7 Wang21 8 Kyoko21 9 ahuai211 1 Toshikazu A T B 2 A B 3 λ 1 λ 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 η η 8

3 k q 2 Δ - Π t = V t - ΔS t V S t t + dt dπ = dv - ΔdS = V V + μs t S V ( 2 σ2 S 2 + q) dt + σs V 2 S S dw t - ΔμSdt + σsdw t Δ = V S dπ = V t V ( 2 σ2 S 2 + q) dt 2 S t t + dt t t + dt λ 1 dt 7 1 R R 1 R R 2 ks1 - η 1 η 1 Π dπ = 1 - λ 1 dt V t 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 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 V 2 σ2 S 2 S - λ rv + λ 1 max ks1 - η 1 + q = V t + r + λ1η S V 1 S V 2 σ2 S 2 S - λ 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

4 ks1 - η 2 η 2 > η 1 η 2 Π ( ) dπ = 1 - λ 1 dt V t V 2 σ2 S 2 + q dt + λ S 2 1 dtδsη 2 - V+ λ 1 dtmaxr ks1 - η λ 1 dt V t 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 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 > 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 e -λ1 +λ2t > T P 3 = P 2 > T= e -λ 2 T x = S = T - t 3 Cauchy V - ( r + λ 1η 1 - ) σ2 V 2 x V 2 σ2 x + λ rv - λ 1 max ke x 1 - η 1 - q = V = = maxke x V = Ue α + βx α β U U 2 σ2 x - e -α -βx λ 2 1 max ke x 1 - η 1 + q= 8 Ux = e -βx maxke x

5 2 121 ( 1 ) α = - λ 1 - r σ 2 2 σ2 - r - λ 1 η 2β = r σ - λ 1η 1 2 σ 2 8 U Ux = e -βx maxke x Ux = σ 槡 2π - e -x-ξ 2σ2 -βξ maxke ξ dξ 1 U U 2 σ2 - e -α -βx λ x 2 1 max ke x 1 - η 1 + q= Ux = Ux = σ 槡 2π λ 1 max ke ξ 1 - η 1 + q e - x-ξ2 2σ2 -ζ -αζ-βξ dξdζ - 槡 - ζ Ux = e -x-ξ σ 槡 2π 2σ2 -βξ maxke ξ dξ 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

6 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 +

7 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 λ 2 3 λ 1 = 5 λ 2 λ 2 4 λ 1 = 3 λ 2 λ 2 5 r 5 r λ 1

8 λ 1 3 λ 1 = 5 λ 2 4 λ 1 = 3 λ 2 5 r 6 σ 6 σ 6 σ 7 R

9 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 BRENNAN M J SCHWARTZ E S Convertible bondsvaluation and optimal strategies for call and conversion J Journal of inance BRENNAN M J SCHWART E S Analyzing convertible bondsj Journal of inance and Quantitative Analysis BARONE 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 KIMURA T SHINOHARA T Monte carlo analysis of convertible bonds with reset clauses J European Journal of Operational Research TSIVERIOTIS K ERNANDES C Valuing convertible bonds with credit riskj Journal of ixed Income AYACHE E ORSYTH P A VETZAL K R The valuation of convertible bonds with credit risk J The Journal of Derivatives WANG L L BIAN B J Pricing of perpetual convertible bonds with credit risk under framework of reduce form J Journal

10 of Tongji University KYOKO Y KATSUSHIGE S The valuation of callable-puttable reverse convertible bonds J Asia-Pacific Journal of Operational Research ZHOU Y YI H A free boundary problem arising from pricing convertible bond J Applicable Analysis J J J 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

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