﻿ 多资产期权确定最佳实施边界问题的研究

# 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

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

(s1(t),...,sm(t))=(θ1eω1(T-t),...,θmeωm(T-t)) 。证眀了：当任意t∈(0，T) ，数学模型

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

(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

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:

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