0 引言

Markowitz提出均值-方差模型为现代套期保值理论奠定了基础, 并吸引了大量的学者对此进行推广和研究.文献[1]研究了资产价格为特殊半鞅时在随机利率下运用均值-方差通过适当的概率测度变换, 将具有随机利率的情形简化为非随机利率情形, 再利用Galtchouk-Kunita-Watanabe分解, 获得了资产价格为一般的特殊半鞅具有随机利率的均值-方差套期保值策略.文献[2]在标的资产价格服从具有随机方差的几何布朗运动, 且随机方差服从一个具有最大波动幅度的几何布朗运动时, 在均值-方差准则下, 运用随机微分对策的方法给出了期权的最优套期保值策略.文献[3]通过概率测度变化和K-W投影技术得到均值-方差准则下的最优套期保值策略.文献[4]研究了股价服从受控的马氏过程时, 在随机市场系数的金融市场中, 先引入倒向随机里卡提方程, 然后运用随机LQ控制得到均值-方差准则下的最优套期保值策略.文献[5]研究了当股价服从受控的马氏过程时, 在随机市场参数的不完备金融市场下, 运用随机LQ控制与倒向随机微分方程的方法在均值-方差准则下给出了最优套期保值策略的显式表示.文献[6]运用动态规划原理, 在标的资产服从由布朗运动和违约过程共同作用下, 把均值-方差准则下的套期保值策略的存在性问题转化为一列耦合倒向随机微分方程解的存在性问题, 得到了最优套期保值策略.文献[7]在保险债务服从重随机Poisson过程时, 采用HJB方法得到了时间一致性均值-方差准则下的寿命风险的最优套期保值策略.文献[8]在股票价格服从跳-扩散过程时, 运用倒向随机微分方程及随机控制理论得到了均值-方差准则下的最优套期保值策略.文献[9-11]在股票价格服从带有Markov调制参数的跳跃-扩散过程时, 通过构造倒向微分方程和随机LQ最优控制方法, 得到了在两个混合未定权益下的最优套期保值策略的显式表示.文献[12-16]研究得到了基于均值方差准则下的最优套期保值策略.以上研究均为单个或两个未定权益在不同准则下的最优套期保值策略, 本文研究了在股票价格服从Levy过程时多个混合未定权益的最优套期保值策略, 并讨论了多个混合未定权益最优套期保值策略与单个未定权益最优套期保值策略的关系.

1 问题框架

假设金融市场上仅有2种证券, 一种是无风险资产, 称为债券P₀, 另一种是风险资产, 称为股票P, 当股票价格受到多种冲击时, 可认为股票价格服从Levy过程.设无风险资产P₀(t) 服从微分方程

$ {\rm{d}}{P_0}\left( t \right) = r{P_0}\left( t \right){\rm{d}}t,{p_o}\left( 0 \right) = 1,t \in \left[ {0,T} \right]. $

(1)

风险资产P(t) 服从微分方程

$ \begin{array}{l} {\rm{d}}P\left( t \right) = P\left( t \right)\left[ {\hat \mu \left( {t,s\left( t \right)} \right){\rm{d}}t + \sigma \left( {t,s\left( t \right)} \right){\rm{d}}W\left( t \right) + \varphi \left( {t,s\left( t \right)} \right)\int_{R/\left\{ 0 \right\}} {\left( {\exp \left( z \right) - 1} \right)\tilde N\left( {{\rm{d}}t,{\rm{d}}z} \right)} } \right],\\ \;\;\;\;\;\;\;\;\;\;\;\;P\left( 0 \right) = P,t \in \left[ {0,T} \right]. \end{array} $

(2)

其中, $\begin{align} & \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mu }\left( t, s\left( t \right) \right)=\mu \left( t, s\left( t \right) \right)+\frac{1}{2}{{\sigma }^{2}}\left( t, s\left( t \right) \right)+ \\ & \int_{R\backslash \left\{ 0 \right\}}{\left( \exp \left( z \right)-1-z{{I}_{\left\{ \left| z \right|<1 \right\}}} \right)}\text{ }\!\!\pi\!\!\text{ }\left( \text{d}z \right) \\ \end{align}$, 设(Ω, F, P) 为一完备的概率空间, W(t) 为(Ω, F, P) 上标准的Brown运动, Y_k为独立同分布列.N(dt, dz) 是R₊×R_-值的Poisson过程, 具有强度dt×π(dz), π(dz) 是R\{0}上的σ-有限Borel测度且$\int_{R\backslash \left\{ 0 \right\}}{\min \left( 1-{{z}^{2}} \right)}\text{ }\!\!\pi\!\!\text{ }\left( \text{d}z \right)<\infty $, 对某个R∈[0, ∞],

$ \tilde N\left( {{\rm{d}}t,{\rm{d}}z} \right) = \left\{ \begin{array}{l} N\left( {{\rm{d}}t,{\rm{d}}z} \right) - {\rm{ \mathsf{ π} }}\left( {{\rm{d}}z} \right){\rm{d}}t,\left| z \right| < 1,\\ N\left( {{\rm{d}}t,{\rm{d}}z} \right),\left| z \right| \ge 1. \end{array} \right. $

W_t与N(dt, dz) 独立.

$ \begin{array}{l} {p_{ij}}\left( t \right) = p\left\{ {s\left( t \right) = j\left| {s\left( 0 \right) = i} \right.} \right\},t > 0,i,j = 1,2, \cdots ,m.\\ {q_{ij}} = \left\{ \begin{array}{l} \mathop {\lim }\limits_{t \to {0^ + }} \frac{{{p_{ij\left( t \right)}}}}{t},i \ne j\\ \mathop {\lim }\limits_{t \to {0^ + }} \frac{{{p_{ij\left( t \right) - 1}}}}{t},i = j \end{array} \right.\;\;\;\;i,j = 1,2, \cdots ,l.\\ {F_t} = \sigma \left\{ {W\left( s \right),N\left( s \right),s\left( s \right):0 \le s \le t} \right\}. \end{array} $

用L_Ft² (T:R) 表示适应F_t的可测随机过程f(t) 而且满足:$\text{E}\int_{0}^{T}{{{\left| f\left( t \right) \right|}^{2}}\text{d}t}<\infty $的集合. L_Ft² (Ω:R) 表示F_t适应的均方可积随机变量ξ的集合, 而且具有范数:‖ξ‖=[E[ξ]²]^1/2.金融市场参数μ(t, s(t)), σ(t, s(t)), φ(t, s(t)) 均为F_t适应的随机过程.

假设投资者拥有的初始财富为x, 风险资产在t时刻的投资量为π(t), 财富为x(t), 那么x(t) 满足下列随机微分方程

$ \begin{array}{*{20}{c}} {{\rm{d}}x\left( t \right) = \left\{ {rx\left( t \right) + \left[ {\left( {\hat \mu \left( {t,s\left( t \right) - r} \right){\rm{ \mathsf{ π} }}\left( t \right)} \right.} \right]} \right\}{\rm{d}}t + {\rm{ \mathsf{ π} }}\left( t \right)\sigma \left( {t,s\left( t \right)} \right){\rm{d}}W\left( t \right) + }\\ {{\rm{ \mathsf{ π} }}\left( t \right)\varphi \left( {t,s\left( t \right)} \right)\int_{R/\left\{ 0 \right\}} {\left( {\exp \left( z \right) - 1} \right)\tilde N\left( {{\rm{d}}t,{\rm{d}}z} \right)} .} \end{array} $

(3)

其中, $\begin{align} & \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mu }\left( t, s\left( t \right) \right)=\mu \left( t, s\left( t \right) \right)+\frac{1}{2}{{\sigma }^{2}}\left( t, s\left( t \right) \right)+ \\ & \int_{R\backslash \left\{ 0 \right\}}{\left( \exp \left( z \right)-1-z{{I}_{\left\{ \left| z \right|<1 \right\}}} \right)}\text{ }\!\!\pi\!\!\text{ }\left( \text{d}z \right) \\ \end{align}$.令$b\left( t, s\left( t \right) \right)=\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\mu }\left( t, s\left( t \right) \right)-r$, 则式(3) 可表示为

$ \begin{array}{*{20}{c}} {{\rm{d}}x\left( t \right) = \left[ {rx\left( t \right) + b\left( {t,s\left( t \right)} \right){\rm{ \mathsf{ π} }}\left( t \right)} \right]{\rm{d}}t + {\rm{ \mathsf{ π} }}\left( t \right)\sigma \left( {t,s\left( t \right)} \right){\rm{d}}W\left( t \right) + }\\ {{\rm{ \mathsf{ π} }}\left( t \right)\varphi \left( {t,s\left( t \right)} \right)\int_{R/\left\{ 0 \right\}} {\left( {\exp \left( z \right) - 1} \right)\tilde N\left( {{\rm{d}}t,{\rm{d}}z} \right)} .} \end{array} $

(4)

定义1  π(t) 是可容许策略, 即π(t) 为使得方程(4) 存在唯一解且π(t)∈L²(T:R).记所有可容许策略集合为U_π={π(t)∈L²(T:R)|方程(4) 存在唯一解}.

定义2  称ξ为一未定权益, 若ξ为F_t可测的, 而且ξ∈L²(Ω:R).

定义3  称ξ为一个混合型未定权益, 若ξ₁, ξ₂, …, ξ_n为n个不同的未定权益而且∀a₁, a₂, …, a_n∈R, a₁, a₂, …, a_n>0, a₁+a₂+…+a_n=1, ξ=a₁ξ₁+a₂ξ₂+…+a_nξ_n.

考虑均值-方差准则下的套期保值问题(P):

$ \left\{ \begin{array}{l} \min {\rm{E}}{\left[ {x\left( T \right) - \left( {{a_1}{\xi _1} + {a_2}{\xi _2} + \cdots + {a_n}{\xi _n}} \right)} \right]^2},\\ s,{\rm{t}}.{\rm{ \mathsf{ π} }}\left( t \right) \in {U_{\rm{ \mathsf{ π} }}},\\ x\left( t \right)满足\left( 1 \right). \end{array} \right. $

2 最优套期保值策略

假设1

$ {\rm{E}}\int_0^T {\left\{ {P\left( t \right)\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]{\rm{ \mathsf{ π} }}\left( t \right)\varphi \left( {t,s\left( t \right)} \right)} \right\}} {\rm{d}}t < \infty . $

假设2   EP²(t) dt < ∞.

假设3   E[π(t)φ(t, s(t))-1]² < ∞.

定理1  在假设1, 2, 3条件下, 问题(P) 的最优解为

$ \begin{array}{l} \pi \left( t \right) = \frac{{\left[ {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right) - x\left( t \right)} \right]}}{{{\sigma ^2}\left( {t,s\left( t \right)} \right)}} \cdot \left[ {\left( {b\left( {t,s\left( t \right)} \right)} \right.} \right. + \\ \;\;\;\;\;\;\;\;\;\;\left. {\varphi \left( {t,s\left( t \right)} \right) + \left. {\frac{{\sigma \left( {t,s\left( t \right)} \right){h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right) - \sigma \left( {t,s\left( t \right)} \right)} \right]. \end{array} $

证明  引入倒向随机微分方程

$ \begin{array}{l} \left\{ \begin{array}{l} {\rm{d}}p\left( {t,s\left( t \right)} \right) = m\left( {t,s\left( t \right)} \right){\rm{d}}t + {h_1}\left( {t,s\left( t \right)} \right){\rm{d}}W\left( t \right),\\ p\left( {T,s\left( T \right)} \right) = 1,p\left( {t,s\left( t \right)} \right) > 0. \end{array} \right.\\ \left\{ \begin{array}{l} {\rm{d}}{\mathit{h}_1}\left( {t,s\left( t \right)} \right) = {\beta _1}\left( {t,s\left( t \right)} \right){\rm{d}}t + {\rm{d}}W\left( t \right),\\ {h_1}\left( {T,s\left( T \right)} \right) = {\xi _1}. \end{array} \right.\\ \left\{ \begin{array}{l} {\rm{d}}{\mathit{h}_2}\left( {t,s\left( t \right)} \right) = {\beta _2}\left( {t,s\left( t \right)} \right){\rm{d}}t + {\rm{d}}W\left( t \right),\\ {h_2}\left( {T,s\left( T \right)} \right) = {\xi _2}. \end{array} \right.\\ \ldots \\ \left\{ \begin{array}{l} {\rm{d}}{\mathit{h}_n}\left( {t,s\left( t \right)} \right) = {\beta _n}\left( {t,s\left( t \right)} \right){\rm{d}}t + {\rm{d}}W\left( t \right),\\ {h_n}\left( {T,s\left( T \right)} \right) = {\xi _n}. \end{array} \right. \end{array} $

$ \begin{array}{l} {\rm{d}}\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right] = \\ \left[ {rx\left( t \right) + b\left( {t,s\left( t \right)} \right){\rm{ \mathsf{ π} }}\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]{\rm{d}}t + \\ \left[ {{\rm{ \mathsf{ π} }}\left( t \right)\sigma \left( {t,s\left( t \right)} \right) - 1} \right]{\rm{d}}W\left( t \right) + \left[ {{\rm{ \mathsf{ π} }}\left( t \right)\varphi \left( {t,s\left( t \right)} \right)} \right]\int_{R/\left\{ 0 \right\}} {\left( {\exp \left( z \right) - 1} \right)\tilde N\left( {{\rm{d}}t,{\rm{d}}z} \right)} . \end{array} $

(5)

运用Ito公式得

$ \begin{array}{l} {\rm{d}}\left\{ {p\left( {t,s\left( t \right)} \right){{\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]}^2}} \right\} = \\ p\left( {t,s\left( t \right)} \right){\rm{d}}{\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]^2} + \\ {\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]^2}{\rm{d}}p\left( {t,s\left( t \right)} \right) + \\ {\rm{d}}p\left( {t,s\left( t \right)} \right) \cdot {\rm{d}}{\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right.} \right]^2} = \\ {\rm{2}}p\left( {t,s\left( t \right)} \right)\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + } \right.} \right.\\ \left. {\left. {{a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]\left( {{\rm{ \mathsf{ π} }}\left( t \right)\sigma \left( {t,s\left( t \right)} \right) - 1} \right){\rm{d}}W\left( t \right) + \\ 2p\left( {t,s\left( t \right)} \right)\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + } \right.} \right.\\ {a_n}{h_n}\left( {t,s\left( t \right)} \right){\rm{ \mathsf{ π} }}\left( t \right)\varphi \left( {t,s\left( t \right)} \right)\int_{R/\left\{ 0 \right\}} {\left( {\exp \left( z \right) - 1} \right)\tilde N\left( {{\rm{d}}t,{\rm{d}}z} \right)} + \\ 2p\left( {t,s\left( t \right)} \right)\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]\left\{ {rx\left( t \right)} \right. + \\ b\left( {t,s\left( t \right)} \right){\rm{ \mathsf{ π} }}\left( t \right) - \left( {{a_1}{\beta _1}\left( {t,s\left( t \right)} \right) + {a_2}{\beta _2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{\beta _n}\left( {t,s\left( t \right)} \right)} \right) + {\rm{ \mathsf{ π} }}\left( t \right)\varphi \left( {t,s\left( t \right)} \right) + \\ \left. {p\left( {t,s\left( t \right)} \right){{\left[ {{\rm{ \mathsf{ π} }}\left( t \right)\sigma \left( {t,s\left( t \right)} \right) - 1} \right]}^2}} \right\}{\rm{d}}t + \\ \sum\limits_{j = 1}^m {{q_{s\left( i \right)j}}\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]} p\left( {t,s\left( t \right)} \right){\rm{d}}t + \\ {\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]^2}m\left( {t,s\left( t \right)} \right){\rm{d}}t + \\ {\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]^2}{\beta _1}\left( {t,s\left( t \right)} \right){\rm{d}}W\left( t \right) + \\ 2\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right] \cdot \\ \left( {{\rm{ \mathsf{ π} }}\left( t \right)\sigma \left( {t,s\left( t \right)} \right) - 1} \right){\beta _1}\left( {t,s\left( t \right)} \right){\rm{d}}t. \end{array} $

(6)

对式(6) 从0到T积分并求数学期望可得

$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{E}}\left\{ {p\left( {T,s\left( T \right)} \right){{\left[ {x\left( T \right) - \left( {{a_1}{h_1}\left( {T,s\left( T \right)} \right)} \right) + {a_2}{h_2}\left( {T,s\left( T \right)} \right) + \cdots + {a_n}{h_n}\left( {T,s\left( T \right)} \right)} \right]}^2}} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;p\left( {0,s\left( 0 \right)} \right){\left[ {x\left( 0 \right) - \left( {{a_1}{h_1}\left( {0,s\left( 0 \right)} \right)} \right. + {a_2}{h_2}\left( {0,s\left( 0 \right)} \right) + \cdots + {a_n}{h_n}\left( {0,s\left( 0 \right)} \right)} \right]^2} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{E}}\int_0^T {p\left( {t,s\left( t \right)} \right)H\left( {t,s\left( t \right)} \right){\rm{d}}t} .\\ H\left( {t,s\left( t \right)} \right) = 2\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + } \right.} \right.\\ \left. {\left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]\left[ {rx\left( t \right) + b\left( {t,s\left( t \right)} \right){\rm{ \mathsf{ π} }}\left( t \right)} \right. - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left( {{a_1}{\beta _1}\left( {t,s\left( t \right)} \right) + {a_2}{\beta _2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{\beta _n}\left( {t,s\left( t \right)} \right)} \right) + {\rm{ \mathsf{ π} }}\left( t \right)\varphi \left( {t,s\left( t \right)} \right)} \right]{\rm{d}}t + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left[ {{\rm{ \mathsf{ π} }}\left( t \right)\sigma \left( {t,s\left( t \right)} \right) - 1} \right]^2} + \sum\limits_{j = 1}^m {{q_{s\left( i \right)j}}\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right)} \right.} \right.} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right] + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + } \right.} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {{a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]\left( {{\rm{ \mathsf{ π} }}\left( t \right)\sigma \left( {t,s\left( t \right)} \right) - 1} \right)\frac{{{h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]^2}\frac{{m\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\sigma ^2}\left( {t,s\left( t \right)} \right){{\rm{ \mathsf{ π} }}^2}\left( t \right) + \left\{ {2\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + } \right.} \right.} \right.\\ \left. {\left. {\left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right] \cdot \left[ {b\left( {t,s\left( t \right)} \right) + \varphi \left( {t,s\left( t \right)} \right) + \frac{{\sigma \left( {t,s\left( t \right)} \right){h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right] \cdot 2\sigma \left( {t,s\left( t \right)} \right)} \right\}{\rm{ \mathsf{ π} }}\left( t \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right] \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {rx\left( t \right) - \left( {{a_1}{\beta _1}\left( {t,s\left( t \right)} \right) + {a_2}{\beta _2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{\beta _n}\left( {t,s\left( t \right)} \right)} \right) - \frac{{h\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right] + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^m {{q_{s\left( i \right)j}}\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]} + 1 + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {t,s\left( t \right)} \right) + {a_2}{h_2}\left( {t,s\left( t \right)} \right) + \cdots + {a_n}{h_n}\left( {t,s\left( t \right)} \right)} \right)} \right]^2}\frac{{m\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}. \end{array} $

(7)

在式(7) 中取

$ \begin{array}{l} {\beta _i}\left( {t,s\left( t \right)} \right) = \frac{{{{\left[ {b\left( {t,s\left( t \right)} \right) + \varphi \left( {t,s\left( t \right)} \right) + \frac{{\sigma \left( {t,s\left( t \right)} \right){h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right]}^2}}}{{ - 2{a_i}{\sigma ^2}\left( {t,s\left( t \right)} \right)}}x\left( t \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{{\left[ {b\left( {t,s\left( t \right)} \right) + \varphi \left( {t,s\left( t \right)} \right) + \frac{{\sigma \left( {t,s\left( t \right)} \right){h_i}\left( {t,s\left( t \right)} \right)}}{{S\left( {t,s\left( t \right)} \right)}}} \right]}^2}}}{{2{\sigma ^2}\left( {t,s\left( t \right)} \right)}}{h_i}\left( {t,s\left( t \right)} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{{\left[ {x\left( t \right) - {a_i}{h_i}\left( {t,s\left( t \right)} \right)} \right]}^2}}}{{2{\sigma ^2}\left( {t,s\left( t \right)} \right)}} \cdot \frac{{m\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}},\left( {1 \le i \le n - 1} \right). \end{array} $

(8)

$ \begin{array}{l} {\beta _n}\left( {t,s\left( t \right)} \right) = \frac{{\left[ {b\left( {t,s\left( t \right)} \right) + \varphi \left( {t,s\left( t \right)} \right) + \frac{{\sigma \left( {t,s\left( t \right)} \right){h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right]}}{{ - 2{a_n}\sigma \left( {t,s\left( t \right)} \right)}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{{\left[ {b\left( {t,s\left( t \right)} \right) + \varphi \left( {t,s\left( t \right)} \right) + \frac{{\sigma \left( {t,s\left( t \right)} \right){h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right]}^2}}}{{2{\sigma ^2}\left( {t,s\left( t \right)} \right)}}{h_n}\left( {t,s\left( t \right)} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2r - \frac{{{h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}} + \sum\limits_{j = 1}^m {{q_{s\left( i \right)j}}} . \end{array} $

(9)

$ \begin{array}{l} {\rm{E}}\left\{ {P\left( {T,s\left( T \right)} \right){{\left[ {x\left( t \right) - \left( {{a_1}{h_1}\left( {T,s\left( T \right)} \right) + {a_2}{h_2}\left( {T,s\left( T \right)} \right) + \cdots + {a_n}{h_n}\left( {T,s\left( T \right)} \right)} \right)} \right]}^2}} \right\} = \\ P\left( {0,s\left( 0 \right)} \right){\left[ {x\left( 0 \right) - \left( {{a_1}{h_1}\left( {0,s\left( 0 \right)} \right) + {a_2}{h_2}\left( {0,s\left( 0 \right)} \right) + \cdots + {a_n}{h_n}\left( {0,s\left( 0 \right)} \right)} \right)} \right]^2} + \\ {\rm{E}}p\left( {t,s\left( t \right)} \right)H\left( {t,s\left( t \right)} \right){\rm{d}}t. \end{array} $

(10)

当β₁(t, s(t)), β₂(t, s(t)), …, β_n(t, s(t)) 分别满足式(8) 和(9) 时, H(t, s(t)) 就为π(t) 的完全平方式.这时π(t) 满足式(10), H(t, s(t))=0, 显然最小.

在式(10) 中, 当a₁=1, a₂=0, …, a_n=0时，可得π_a1(t)；当a₁=0, a₂=1, …, a_n=0时, 可得π_a2(t)；…；当a₁=0, a₂=0, …, a_n=1时，可得π_an(t) 而且有

$ {a_1}{{\rm{ \mathsf{ π} }}_{a1}}\left( t \right) + {a_2}{{\rm{ \mathsf{ π} }}_{a2}}\left( t \right) + \cdots + {a_n}{{\rm{ \mathsf{ π} }}_{an}}\left( t \right) = {\rm{ \mathsf{ π} }}\left( t \right),\forall {a_1},{a_2}, \cdots ,{a_n} > 0,{a_1} + {a_2} + \cdots + {a_n} = 1. $

当a₁=1, a₂=0, …, a_n=0时

$ {\rm{ \mathsf{ π} }}\left( t \right) = {{\rm{ \mathsf{ π} }}_{a1}}\left( t \right) = \frac{{\left[ {{h_1}\left( {t,s\left( t \right)} \right) - x\left( t \right)} \right]\left[ {b\left( {t,s\left( t \right)} \right) + \varphi \left( {t,s\left( t \right)} \right) + \frac{{\sigma \left( {t,s\left( t \right)} \right){h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right] - \sigma \left( {t,s\left( t \right)} \right)}}{{{\sigma ^2}\left( {t,s\left( t \right)} \right)}}. $

当a₁=0, a₂=1, …, a_n=0时,

$ {\rm{ \mathsf{ π} }}\left( t \right) = {{\rm{ \mathsf{ π} }}_{a2}}\left( t \right) = \frac{{\left[ {{h_2}\left( {t,s\left( t \right)} \right) - x\left( t \right)} \right]\left[ {b\left( {t,s\left( t \right)} \right) + \varphi \left( {t,s\left( t \right)} \right) + \frac{{\sigma \left( {t,s\left( t \right)} \right){h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right] - \sigma \left( {t,s\left( t \right)} \right)}}{{{\sigma ^2}\left( {t,s\left( t \right)} \right)}}. $

当a₁=0, a₂=0, …, a_n=1时,

$ \begin{array}{*{20}{c}} {{\rm{ \mathsf{ π} }}\left( t \right) = {{\rm{ \mathsf{ π} }}_{an}}\left( t \right) = \frac{{\left[ {{h_n}\left( {t,s\left( t \right)} \right) - x\left( t \right)} \right]\left[ {b\left( {t,s\left( t \right)} \right) + \varphi \left( {t,s\left( t \right)} \right) + \frac{{\sigma \left( {t,s\left( t \right)} \right){h_1}\left( {t,s\left( t \right)} \right)}}{{p\left( {t,s\left( t \right)} \right)}}} \right] - \sigma \left( {t,s\left( t \right)} \right)}}{{{\sigma ^2}\left( {t,s\left( t \right)} \right)}}.}\\ {{a_1}{{\rm{ \mathsf{ π} }}_{a1}}\left( t \right) + {a_2}{{\rm{ \mathsf{ π} }}_{a2}}\left( t \right) + \cdots + {a_n}{{\rm{ \mathsf{ π} }}_{an}}\left( t \right) = {\rm{ \mathsf{ π} }}\left( t \right).} \end{array} $

即多个未定权益的最优均值-方差套期保值策略相对于单个未定权益具有可加性.

3 结束语

在股票价格服从Levy过程时, 运用随机LQ控制, 并引入倒向随机微分方程得到了均值方差准则下具有多个混合型未定权益的最优套期保值策略, 同时讨论了多个混合未定权益与单个未定权益最优套期保值策略的关系, 即多个混合型未定权益的最优套期保值策略相对于单个未定权益具有可加性，即具有凸性.