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

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)，...，s_m(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),...,s_m(t))=(θ₁eω1^(T-t),...,θ_me^ωm(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∠S_j∠∞，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),...,s_m(t))=(θ₁eω1^(T-t),...,θ_me^ωm(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),...,s_m(t))=(θ₁eω1^(T-t),...,θ_me^ωm(T-t)) 为最佳实施边界。指数函数向量

(s₁(t),...,s_m(t))=(θ₁eω1^(T-t),...,θ_me^ωm(T-t)) 满足条件 ，

k=1，...，m；且有ω_k 的计算公

式 ；公式表明ω_{k，k=1，...，m} 由多维

Black-Scholes方程中出现的所有参数a_kj ，q_j ，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)，...，s_m(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-

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),...,s_m(t))=(θ₁eω1^(T-t),...,θ_me^ωm(T-t))  . It is demonstrated that: when any

t∈(0，T) ,the maximum value of the solution u(s，t) of the region

R₊ ^m：0∠S_j∠∞，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₊ ^m：0∠S_j∠∞, j=(s₁(t),...,s_m(t))=(θ₁eω1^(T-t),...,θ_me^ωm(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),...,s_m(t))=(θ₁eω1^(T-t),...,θ_me^ωm(T-t)) satisfies

the condition ，k=1，...，m; and ω_k is calculated

by; the formula shows that ω_k is only determined by all

the parameters appearing in the multidimensional Black-Scholes equation.