32 Vol 32 2 Journal of Harbin Engineering Univerity Jan 2 doi 3969 /j in 6-743 2 23 5 2 F83 9 A 6-743 2-24-5 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 29-9-28 HEUF422 957- E-mail zhengxiaoyang @ hrbeu edu cn 985- E-mail guanchang@ hrbeu edu cn Black-chole Ingeroll 977 Brennan chwartz 977
25 2-4 4 5 5 6 7 6-8 9 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 2 2 2 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
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 5 6 2 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 } 8 2 2 2 2 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
27 3 2 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
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