标题:
多资产期权确定最佳实施边界问题的研究Research on the Implementation of the Optimal Implementation of the Multi-Asset Option
作者:
吴小庆
关键字:
多资产期权, 最佳实施边界, 自由边界问题, 多维Black-Scholes方程Multi-Asset Option, Best Implementation Boundary, Free Boundary Problem, Multi-Dimension Black-Scholes Equation
期刊名称:
《Pure Mathematics》, Vol.6 No.6, 2016-11-30
摘要:
本文研究多资产期权确定最佳实施边界的问题，建立了多维Black-Scholes方程在多维区域
Ω≅{(s,t)|s∈R+ m，t∈(0，T)} 具有奇异内边界函数向量s=s(t)=(s1(t)，...，sm(t))，
0∠t∠T 的数学模型，期权价格函数为未知函数。应用矩阵理论和广义特征函数法获得了期权价格函
数的精确解 u(s，t)。并获得了奇异内边界的指数函数向量表达式
(s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t)) 。证眀了：当任意t∈(0，T) ，数学模型
的解u(s，t)在奇异内边界取区域R+ m：0∠Sj∠∞，j=1，...，m 中的最大值，即
u(s(t)，t)= t∈(0，T) ；同时获得了 Black-Scholes方程的自由边界问题A和自由
边界问题B的精确解和其自由边界的指数函数向量表达式
(s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t)) ，问题A和问题B的自由边界与奇异内边界
重合。从而指数函数向量表达式
s(t)=(s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t)) 为最佳实施边界。指数函数向量
(s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t))
满足条件 ，
k=1，...，m；且有ωk 的计算公
式 ；公式表明ωk，k=1，...，m 由多维
Black-Scholes方程中出现的所有参数akj ，qj ，r 唯一确定。
In this paper, we study the problem of determining the optimal implementation boundary
of multi- asset option, and establish a mathematicalmodel of multidimensional Black-Scholes
equation with singular inner boundary function vector
s=s(t)=(s1(t)，...，sm(t))，0∠t∠T , In multi-dimension region Ω≅{(s,t)|s∈R+ m，t∈(0，T)}
the option price function is an unknown function. The exact solution u(s，t) of the mathem-
aticalmodel is obtained by using thematrix theory and the generalized characteristic function
method. And the exponential function vector expression of the singular inner boundary is ob-
tained (s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t)) . It is demonstrated that: when any
t∈(0，T) ,the maximumvalue of thesolution u(s，t) of the region
R+ m：0∠Sj∠∞，j=1，...，mis obtained onthe singular boundary, namely u(s(t)，t)= .
The free boundary problem A and free boundary problem B of Black-Scholes equation are solved.
The free boundary of problem A and B is expressed by the function vector
R+ m：0∠Sj∠∞, j=(s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t))1，...，m .
Thefreeboundary of the problem A and problem Bcoincides with the singular inner boundary. So
the vector expression of the exponential function is the best implementation of the boundary. The
exponential function vector (s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t)) satisfies
the condition ，k=1，...，m; and ωk is calculated
by; the formula shows that ωk is only determined by all
the parameters appearing in the multidimensional Black-Scholes equation.