本文建立了Black-Scholes方程在区域Ω:0s∞,0tT具有多条奇异内边界s=sj(t),0tT;j∈{0,1,...,N}的数学模型,引入广义特征函数法获得了数学模型的精确解u(s,t) ,并进一步获得奇异内边界是指数函数曲线sj(t)=sjTeσ²ω(T-t) ,j∈{0,1,...,N} ,证明了在任意时刻t∈(0,T) ,函数u(s,t) 在闭区间[0,so(t)]中的最大值在奇异内边界so(t) 上取得,区间[sN(t),∞] 中的最大值在奇异内边界sN(t)上取得。特别地,考虑在区域Ω内仅有一条奇异内边界s=s(t),0tT的数学模型,获得了奇异内边界是指数函数曲线s(t)=sTeσ²ω(T-t) ,证明了:解在奇异内边界s=s(t),0tT 取最大值,即u(s(t),t)=u(s,t) ;且问题IIIA和IIIB的自由边界与奇异内边界重合,指数函数曲线s(t)=sTeσ²ω(T-t) 就是美式期权最佳实施边界。 In this paper, the mathematical model is established of the Black Scholes equation in the regionΩ:0s∞,0tT with a number of singular inner boundar s=sj(t),0tT;j∈{0,1,...,N}, and introduce the generalized characteristic function method to be able to obtain the exact solution of the mathematical model, and further to obtain singular boundary is exponential function curve sj(t)=sjTeσ²ω(T-t) ,j∈{0,1,...,N} , It is proved that the maximum value of the exact solution u(s,t) in the closed interval [0,so(t)] is on the singular boundary so(t) , the maximum value in the interval [sN(t),∞] is obtained on the singular boundary sN(t) . In particular, consider the mathematical model with only a singular boundary, the maximum value in the interval [0,∞) of solution u(s,t) on the singular boundary s=s(t),0tT that is, u(s(t),t)=u(s,t) . The free boundary problem of IIIA and IIIB about Black Scholes equation are all solved. At the same time to obtain exponential function curve s(t)=sTeσ²ω(T-t) of the free boundary, and singular boundary coincides, so the curve s(t)=sTeσ²ω(T-t) is American option implement best boundary.
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