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 :西南石油大学理学院,四川 成都;

关键词: 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

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+ mt∈(0,T)} 具有奇异内边界函数向量s=s(t)=(s1(t),...,sm(t)),

0∠t∠T 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+ m0∠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

;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。



Abstract:

In this paper, we study the problem of determining the optimal implementation boundary 

of multi- asset option, and establish a mathematical model 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+ mt∈(0,T)}

the option price function is an unknown function. The exact solution u(s,t) of the mathem-

atical model is obtained by using the matrix 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 maximum value of the solution u(s,t) of the region

R+ m0∠Sj,j=1,...,m is obtained on the 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+ m0∠Sj, j=(s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t))1,...,m . 

The free boundary of the problem A and problem B coincides 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.



Abstract:

文章引用: 吴小庆 (2016) 多资产期权确定最佳实施边界问题的研究。 理论数学, 6, 496-526. doi: 10.12677/PM.2016.66068

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