摘要:
本文建立了Black-Scholes方程在区域Ω：0<s<∞，0<t<T具有多条奇异内边界s=sj(t)，0<t<T；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)，0<t<T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)，0<t<T 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：0<s<∞,0<t<T with a number of singular inner boundar s=sj(t),0<t<T;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),0<t<T 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.