The martingale pricing method for pricing fluctuation concerning stock models of callable bonds with random parameters

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1 32 Vol 32 2 Journal of Harbin Engineering Univerity Jan 2 doi 3969 /j in F83 9 A he martingale pricing method for pricing fluctuation concerning tock model of callable bond with random parameter ZHENG Xiaoyang GUAN Chang College of cience Harbin Engineering Univerity Harbin 5 China Abtract In order to tudy the pricing of convertible bond with both bond and option propertie and alo for further application of the martingale method to option pricing a model of tock pricing fluctuation which ha random and abnormal fluctuation and can better decribe the actual market ituation wa adopted hi wa done conidering the background of the financial crii and with the aumption that the rik free rate the expected return rate and the volatility are random time function parameter Furthermore knowledge about tochatic differential theory meaure tranformation and the martingale theory wa applied to analyze the pricing of callable bond Finally the pricing formula for the initial price wa obtained he exiting achievement were further improved to a more practical level he reult ha laid a foundation for further reearch about convertible pricing bond and extended the application of the martingale theory Keyword model of tock pricing fluctuation callable bond random parameter martingale method 973 Black-chole HEUF hrbeu edu cn guanchang@ hrbeu edu cn Black-chole Ingeroll 977 Brennan chwartz 977

2 d t = α t t dt + μ t t dt + σ t t dw p t 2 α t 6 W P t t Ω F P F t t W P t t σ 2 Giranov Giranov Ω F P Ω σ θ t t θ 2 d < t t t = exp - t θ dw P - 2 θ t 2 d 3 W P t t P P t C W Q t = W P t t + P Ω F C C = 2 P < P C < C P C C < 3 r t t 2 Giranov Giranov θ d W Q t t Q E P I A = E Q I A A F Giranov 2 2 P dp dp = exp ( - θ t dw P - 2 θ 2 d ) μ t - r t θ t = σ t θ 2 d < Giranov 2 P P 5 μ t σ t Ω F P 4 W P t = W P t + t θ d 5

3 26 32 Λ = dp dp = exp ( - t dw θ P - 2 θ 2 d) Giranov E p Λ I A = E P I A A F dw P t = dw P t - θ t dt 6 d t = α t t dt + r t t dt + σ t t dw p t 7 = exp{ α + r - 2 σ2 d + σ dw P } P 2 dp 2 = exp dp σ dw P - 2 σ 2 d 9 P 2 P W P 2 t = W P t - t σ d Ω F P 2 Λ 2 = exp σ dw P - 2 σ 2 d E P Λ 2 I A' = E P2 I A' A' F 2 dw P t = dw P 2 t + σ t dt d t = α t t dt + r t t dt + σ t t dw P t 2 = exp{ α + r + 2 σ2 d Y P Y < y = N y - μ 2μy exp{ σ 2 d } N σ 2 d - y - μ σ 2 d - 2 X = ln Y μ X Y x - μ P X < x Y < y = N σ 2 d - 2μy x - 2y - μ exp{ σ 2 d } N σ 2 d 3 C { } C = exp - r d E P C = r d } r d } [ ] ) ] E P P I < P C < + E P r d } C I P C < + E P C I 3 V V 2 V 3 3 V A = r d } V = { - r d } [ ] E P P I < P C < = P AE P ln < ln P C [ ln < ln ] 2 σ2 d + x = ln σ dw P 2 } P C y = ln μ = 3 ( α + r - x - μ 2 σ2 ) d d = 2 σ 2 d 2 Y = ln μ = α + r - V = P AP X < x Y < y = x - μ N σ 2 d - e y 2μ P A x - 2y - μ σ 2 d P A N d - 2μ σ d σ - = σ 2 d

4 V 2 [ ] V 2 = AE P C I P C < = C AE P exp ( α + r - 2 σ2 d ) + σ dw P { } I P C < ) ] = C exp { α d } E P exp σ dw P - 2 σ 2 d I P C < ) ] { } ) ] I P C < ) ] = n = C B = exp α d V 2 = nbe P Λ 2 I P C nbe P2 < = nbp 2 ln ln P C ln < ln = nbp 2 X x Y < y = nb P 2 Y < y - P 2 X < x Y < y P 2 X < x Y < y P X < x Y < y μ = ( α + r + 2 x - μ d ) 2 d 2 = σ 2 d N d 2 - P 2 X < x Y < y = σ d 2 - σ 2 d y - μ 2 d 3 = σ 2 d d 4 = - y - μ 2 σ 2 d P 2 Y < y = N d 3 - V 2 σ2 d N d 4 = nb N d 3 - ( ) { d σ 4 - N d 2 + } σ 2 d 3 3 V 3 V 3 = AE P [ C I ] = nae P exp σ dw P [ { + ( α + r - 2 σ2 d ) } I ] = - 2 σ2 d ) I ] nae P exp σ dw P [ ] nb - P 2 ln < ln = nbe P = nb - P 2 Y < y Λ 2 I = nbe P2 = nb - N d 3 + nb N d 3 - nb - N d 3 + C = P A N d - σ d 4 - N d 2 + σ 2 d [ N d 4 ] nb - N d 2 + σ d - = P A N d - I = σ 2 d [ N d 4 ] + σ 2 d σ d - σ 2 d + σ 2 d + σ 2 d

5 ZHU Dan he martingale pricing for convertible bond with back ell treaty J Journal of Natural cience of Hunan Normal Univerity J ZHU Dan he martingale pricing for convertible bond with CHEN houhong Convertible bond invetment and financing theory and practice hanghai hanghai Univerity of Finance and Economic Pre BACK F CHOE he pricing of option and corporate liabilitie J Journal of Political Economy INGERO J E A contingent-claim valuation of convertible ecuritie J Journal of Financial Economic BRENNAN J CHWARZ E Convertible bond valuation and optimal trategie for call and converion J Journal of Finance B- J ZHU Dan he martingale pricing for convertible bond of WANG Qingliu ZOU Huiwen A urvey of the convertible bond pricing theory and baed on the B- option model' applied reearch J Journal of Harbin Univerity of Commerce ocial cience Edition J ZHU Dan YANG Xiangqun he martingale pricing for convertible bond J tatitic and Deciion J back buy treaty J Journal of Hunan Univerity of cience and Engineering J FU Fenjuan IU Xinping he martingale pricing for convertible bond on tock with dividend J Journal of Hainan Univerity Natural cience J ZHOU Qiyuan WU Chongfeng IU Hailong Valuing the callable convertible dicount bond with credit rik an e- quivalent decompoition method J Journal of hanghai Jiao ong Univerity J the model of tock pricing fluctuation J Finance and E- conomy D FU Fenjuan he tudy on the pricing of everal pecial type of convertible bond D Xian haanxi Normal U- niverity

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DOI /J. 1SSN

DOI /J. 1SSN 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