由于金融市场自身固有的波动性,以及人们对客观市场进 行刻画时的主观误差,许多金融数据不能用确定的数来表示, 例如市场无风险利率和波动率,由于不同的银行之间利率不同, 通常我们会说无风险利率为5\%左右,市场中的波动率 也很难用确切的数描述出来,人们也许会说波动率为3\%左右, 这些不确定的数均具有模糊性.为了描述模糊概念与模糊 现象,扎德(Zadeh)^{[1, 2, 3]}提出了模糊集合(fuzzy sets)的概念, 把普通集合中的绝对隶属关系加以扩 充,使得元素与``集合"的隶属度由只能取0和1这两个值推广到单位区间 中的任意的数值,从而实现定量地刻画模糊 事物.可见,将模糊理论引入到期权定价中是对定价理论的很好的补充, 而且,近年来许多学者对模糊期权定价理论 做了大量的研究.

Yoshida^[4]在B-S模型中,假定股票价格为对称的三角形模糊数, 并且假定模糊程度与股票价格成比例,定义 模糊测度下的模糊期望,求出欧式看涨与看跌期权的模糊区间, 以及对冲策略,最后给出模糊环境下的欧式看涨期权 与看跌期权的平价公式.Yoshida^[5]研究了B-S模型下的美 式期权定价问题,引入模糊随机变量,假定股票价 格为三角模糊数,求得美式期权价格区间,以及对冲策略. 但是,以上两篇文章中的目标函数的选取因人而异,从而不利 于统一定价.此外,该模型只考虑了股票价格这一个模糊因素, 而其它影响期权价格的具有模糊性的因素未考虑,如无风险利率、波动率等.

台湾学者Wu^{[6, 7, 8]}将模糊算术引入到B-S公式中,并对B-S公式的模 糊形式进行不同的分析 得到期权定价公式.其中2007年的文章中假定市场中的利率,波动率, 股票价格都是模糊的,利用模糊数学的扩展原理, 由B-S模型下的欧式期权的定价公式诱导出模糊环境中期权定价公式, 通过敏感度分析求出期权价格的模糊区间,进而 得到模糊期权的隶属度.Thiagarajah^[9]在考虑支付红利函数下, 假定利率、波动率,股票价格为一种改进的模 糊数,推导出期权的模糊价格区间.

以上的文献中均未讨论模糊环境中交易费用问题.蹇明^[10]在B-S模 型的基础上考虑交易费用,建立了带有交 易费用的定价模型,并且假定交易费用率为模糊数,得到权证价格的模 糊区间的端点满足的偏微分方程.最后选取证 券市场上两支隶属于不同行业板块-石化CWB1和青啤CWB1 -的权证进行实证分析,与Leland的交易费用模型相比更加符 合市场情况.但是该文的研究是基于期权价格与交易费用成正比的假设下完成的.

此外,市场存在跳现象,Xu^[11]研究了模糊环境中跳扩散模型的 期权定价问题.假设市场利率、波动率、跳 跃强度、跳跃幅度为三角模糊数,利用模糊理论通过等价鞅测度方法 得到欧式看涨期权的定价公式,利用模糊算法得 到期权的模糊区间.Nowak和Romaniuk^[12]研究了模糊环境中, 当标的资产服从列维过程时的期权定价方法. 由于在不完全市场中,与市场测度等价的鞅测度并不是唯一的, 文中选取其中最小商鞅测度利用鞅方法给出欧式看涨 期权的定价公式以及模糊定价公式,并利用蒙特卡洛模拟得到数值试验.

由于市场中同时存在跳现象和交易费用,本文将讨论模糊环境 中跳扩散模型下带有交易费用期权定价方法.

在导出本文的模型之前,先给出一些基本假设,并假定本文第3和4部 分的模型都以此为假设基础.

假设标的资产股票价格由与时间有关的连续部分和跳跃部分构成,用下面的 微分方程表述出来

$$ {\rm d}S_t = S_{t^ - } \left[{\left( {\mu - \lambda m} \right){\rm d}t + \sigma {\rm d}Z_t + U{\rm d}N_t } \right] $$

(2.1)

其中$Z_t$是一个标准的Brown运动,$\mu$是股票的瞬时期望收益率, $\sigma$为股票不发生跳跃时候的收益方差(即泊松跳不发生); $N_t $是强度为 $\lambda $且与 $Z_t$相互独立的泊松跳过程, $U$是跳跃发生时跳跃百分比的随机变量,且满足: $m = E(U)$, $\ln (1 + U)$服从正态分布,其方差为$\theta ^2 $,也即: $\ln (1 + U) \sim N(\ln (1 + m) - \frac{1}{2}\theta ^2 , \theta ^2 )$; $E^U $是随机变量$U$的期望.

考虑市场中存在的交易费用作如下假设:

$\bullet$~ 投资组合头寸每隔$\Delta t$ 时间段调整一次, 并且$\Delta t$相对于$T$是一个固定、非无穷小的数.例如: 有效期为一年的期权,投资组合头寸每周调整一次;

$\bullet$~ 股票的交易费用与股票交易价格成一定比例$\kappa $,用$\left| \nu \right|$ 表示股票交易的份额,$\nu > 0$ 表示买入股票,$\nu < 0$表示卖出股票, 这是因为不管是买入还是卖出都需要支付交易费用. 因此,需要支付的交易费用为$\kappa \left| \nu \right|S$;

$\bullet$~ 假设在$\Delta t$ 时间内,市场中不发生重大的经济变化, 股票的变动是由于公司或企业本身的经营造成的, 因此跳跃部分的风险是``非系统风险".

下面不妨以欧式看涨期权为例,在给出跳扩散模型下带有交易费 用的欧式看涨期权定价公式前,我们先引入以下引理.

引理3.1     (姜礼尚^[13]) 股票价格服从Merton 跳扩散模型(2.1)时,其解析解为

\begin{equation} S_t = S_0 \exp \left\{ {\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)t + \sigma W_t + \sum\limits_{t = 0}^{N_t } {\ln \left( {1 + U_i } \right)} } \right\} , %(3.1) $$ \end{equation}

(3.1)

对应的,以此股票为标的资产的欧式看涨期权公式如下

\begin{equation} C\left( {S_t ,t} \right) = \sum\limits_{n = 0}^\infty {\frac{{\left[{\lambda \left( {1 + m} \right) \left( {T - t} \right)} \right]^n }}{{n!}}} S_t {\rm e}^{ - \lambda \left( {1 + m} \right)\left( {T - t} \right)} [S_t {\rm{(}}d_2 {\rm{)}} - K{\rm e}^{ - r_n (T - t)} N(d_1 ){\rm{]}}, % (3.2) $$ \end{equation}

(3.2)

其中 \[d_1 = \frac{{\ln \frac{{S_t }}{K} + (r_n - \frac{1}{2}\sigma _n ^2 )(T - t)}}{{\sigma _n \sqrt {T - t} }},\qquad d_2 = \frac{{\ln \frac{{S_t }}{K} + (r_n + \frac{1}{2}\sigma _n ^2 )(T - t)}}{{\sigma _n \sqrt {T - t} }}, \] 其中 \[\sigma _n ^2 {\rm{ = }}\sigma ^{\rm{2}} {\rm{ + }}\frac{{n\sigma _u ^2 }}{{T - t}},\qquad r_n = r - \lambda m + \frac{n}{{T - t}}(\mu - \frac{1}{2}\sigma _u ^2 ), \] 其中 \[m = E(U) = \exp(\mu + \frac{{\sigma _u ^2 }}{2}) - 1, ~~ \textrm{也即} \sigma _u ^2 = 2(\ln (m + 1) - \mu ). \]

引理3.2     (Mocioalca^[14]) 带有交 易费用的Merton跳扩散模型的欧式看涨期权满足的偏微分方程(PDE)是 对不考虑 交易费用的定价模型满足的PDE中的波动率做了如下修正

\begin{equation} {\sigma'} ^2 = \sigma ^2 - \frac{2}{{\Delta t}}\kappa E\left( {\left| {\frac{{\Delta S}}{S}} \right|} \right). % (3.3) $$ \end{equation}

(3.3)

定理3.1     跳扩散模型(2.1)下,到期日$T$,执行价格为$K$,交易费用率为$\kappa $的欧式看涨期权的定价公式为 \begin{eqnarray*} C\left( {S_t ,t} \right) &=& \sum\limits_{n = 0}^\infty {\frac{{\left[{\lambda \left( {1 + m} \right)\left( {T - t} \right)} \right]^n }}{{n!}}} S_t {\rm e}^{ - \lambda \left( {1 + m} \right)\left( {T - t} \right)} N\left( {d_1 } \right)\\ && - \sum\limits_{n = 0}^\infty {\frac{{\left[{\lambda \left( {T - t} \right)} \right]^n }}{{n!}}} {\rm e}^{ - \left( {\lambda + r} \right)\left( {T - t} \right)} KN\left( {d_2 } \right), \end{eqnarray*} 其中

\begin{equation} \label {eqn 3.4} \begin{array}{l} \displaystyle d_1 = \frac{{\ln \frac{{S_t \left( {1 + m} \right)^n }}{K} + \left( {r - \lambda m + \frac{1}{2}\sigma ^2 } \right)\left( {T - t} \right) + \frac{1}{2}n\theta ^2 }}{{\sqrt {{\sigma'} ^2 \left( {T - t} \right) + n\theta ^2 } }} ,\\ [6mm] \displaystyle d_2 = d_1 - \sqrt {{\sigma'} ^2 \left( {T - t} \right) + n\theta ^2 }, \end{array} %(3.4) \end{equation}

(3.4)

其中 \[{\sigma'} ^2 = \sigma ^2 - \frac{2}{{\Delta t}}\kappa \sum\limits_{n = 0}^\infty {\frac{{{\rm e}^{ - \lambda \Delta t} \left( {\lambda \Delta t} \right)^n }}{{n!}}\left\{ {\left( {1 + m} \right)^n {\rm e}^{\left( {r - \lambda m} \right)\Delta t} \left[{1 - 2N\left( {l_1 } \right)} \right] - \left[{1 - 2N\left( {l_2 } \right)} \right]} \right\}}, \] 其中 \[l_1 = \frac{{\ln \left( {1 + m} \right)^n + \left( {r - \lambda m + \frac{1}{2}\sigma ^2 } \right)\Delta t}}{{\sqrt {\sigma ^2 \Delta t + n\theta ^2 } }},\qquad{} {\rm{ }}l_2 = l_1 - \sqrt {\sigma ^2 \Delta t + n\theta ^2 } . \]

证     根据引理3.2,为了得到带有交易费用的期权价格, 我们需要计算$E\left( {\left| {\frac{{\Delta S}}{S}} \right|} \right)$ , 这里我们利用(3.1)式得到 \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \Delta S &=& S_{t + \Delta t} - S_t \\ &=& S_0 \exp \left\{ {\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\left( {t + \Delta t} \right) + \sigma W_{t + \Delta t} + \sum\limits_{t = 0}^{N_{t + \Delta t} } {\ln \left( {1 + U_i } \right)} } \right\} \\ & & - S_0 \exp \left\{ {\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)t + \sigma W_t + \sum\limits_{t = 0}^{N_t } {\ln \left( {1 + U_i } \right)} } \right\} \\ &=& \left[{S_0 \exp \left\{ {\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)t + \sigma W_t + \sum\limits_{t = 0}^{N_t } {\ln \left( {1 + U_i } \right)} } \right\}} \right] \\ & & \left[{\exp \left\{ {\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + \sigma \left( {W_{t + \Delta t} - W_t } \right) + \sum\limits_{t = 0}^{N_{\Delta t} } {\ln \left( {1 + U_i } \right)} } \right\} - 1} \right] , \end{eqnarray*} 所以 \[ E\left( {\left| {\frac{{\Delta S}}{S}} \right|} \right) = E\left( {\left| {\exp \left\{ {\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + \sigma \left( {W_{t + \Delta t} - W_t } \right) + \sum\limits_{t = 0}^{N_{\Delta t} } {\ln \left( {1 + U_i } \right)} } \right\} - 1} \right|} \right). \] 记 \[Y = \sigma \left( {W_{t + \Delta t} - W_t } \right) + \sum\limits_{i = 0}^n {\ln \left( {1 + U_i } \right)}, \] 则 \[Y \sim N\left( {n\ln \left( {1 + m} \right) - \frac{1}{2}n\theta ^2 ,\sigma ^2 \Delta t + \frac{1}{2}n\theta ^2 } \right) \buildrel \Delta \over = N\left( {s,v^2 } \right). \] 所以

\begin{eqnarray} E\left( {\left| {\frac{{\Delta S}}{S}} \right|} \right) &=& E\left( {\left| {\exp \left\{ {\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + \sigma \left( {W_{t + \Delta t} - W_t } \right) + \sum\limits_{t = 0}^{N_{\Delta t} } {\ln \left( {1 + U_i } \right)} } \right\} - 1} \right|} \right) \nonumber\\ &= &\sum\limits_{n = 0}^\infty {\frac{{\left( {\lambda \Delta t} \right)^2 {\rm e}^{ - \lambda \Delta t} }}{{n!}}} \int_{ - \infty }^\infty {\left| {\exp \left[{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + y} \right] - 1} \right|} \frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}} {\rm d}y \nonumber \\ &=& \sum\limits_{n = 0}^\infty {\frac{{\left( {\lambda \Delta t} \right)^2 {\rm e}^{ - \lambda \Delta t} }}{{n!}}} \int_{ - \left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t}^\infty {\left\{ {\exp \left[{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + y} \right] - 1} \right\}} \nonumber\\ && \frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}}{\rm d}y - \sum\limits_{n = 0}^\infty {\frac{{\left( {\lambda \Delta t} \right)^2 {\rm e}^{ - \lambda \Delta t} }}{{n!}}} \int_{ - \infty }^{ - \left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} \nonumber \\ &&{\left\{ {\exp \left[{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + y} \right] - 1} \right\}} \frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}} {\rm d}y \nonumber\\ &= &\sum\limits_{n = 0}^\infty {\frac{{\left( {\lambda \Delta t} \right)^2 {\rm e}^{ - \lambda \Delta t} }}{{n!}}} l, \end{eqnarray}

(3.5)

其中 \begin{eqnarray*} % \nonumber to remove numbering (before each equation) J_1 &=& \int_{ - \left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t}^\infty {\left\{ {\exp \left[{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + y} \right] - 1} \right\}} \frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}}{\rm d}y \\ &=& \int_{ - \infty }^{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} {\left\{ {\exp \left[{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + y} \right] - 1} \right\}} \frac{1}{{\sqrt {2\pi } v}}\exp \left[{ - \frac{{\left( { - y - s} \right)^2 }}{{2v^2 }}} \right]{\rm d}y \\ &=& \exp \left\{ {\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} \right\}\int_{ - \infty }^{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} {{\rm e}^{ - y} \frac{1}{{\sqrt {2\pi } v}}\exp \left[{ - \frac{{\left( { - y - s} \right)^2 }}{{2v^2 }}} \right]}{\rm d}y \\ & & - \int_{ - \infty }^{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} {\frac{1}{{\sqrt {2\pi } v}}\exp \left[{ - \frac{{\left( { - y - s} \right)^2 }}{{2v^2 }}} \right]}{\rm d}y \\ &=& \exp \left[{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} \right]\exp \left( {s + \frac{{v^2 }}{2}} \right)N\left( { - \frac{{s + v^2 + (r - \lambda m - \frac{1}{2}\sigma ^2 )\Delta t}}{v}} \right) \\ & & - N\left( { - \frac{{s + (r - \lambda m - \frac{1}{2}\sigma ^2 )\Delta t}}{v}} \right) \\ &=& \left( {1 + m} \right)^n {\rm e}^{\left( {r - \lambda m} \right)\Delta t} N\left( { - l_1 } \right) - N\left( { - l_2 } \right) \end{eqnarray*} \begin{eqnarray*} % \nonumber to remove numbering (before each equation) J_2 &=& \int_{ - \infty }^{ - \left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} {\left\{ {\exp \left[{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + y} \right] - 1} \right\}} \frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}}{\rm d}y \\ &=& \int_{ - \infty }^{ - \left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} {\exp \left[{\left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t + y} \right]\frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}} }{\rm d}y \\ & & - \int_{ - \infty }^{ - (r - \lambda m - \frac{1}{2}\sigma ^2 )\Delta t} {\frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}} }{\rm d}y \\ &=& \exp \left\{ { - \left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} \right\}\int_{ - \infty }^{ - (r - \lambda m - \frac{1}{2}\sigma ^2 )\Delta t} {{\mathop{\rm e}\nolimits} ^y \frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}} }{\rm d}y \\ & & - \int_{ - \infty }^{ - (r - \lambda m - \frac{1}{2}\sigma ^2 )\Delta t} {\frac{1}{{\sqrt {2\pi } v}}{\rm e}^{ - \frac{{\left( {y - s} \right)^2 }}{{2v^2 }}} }{\rm d}y \\ &=& \exp \left\{ { - \left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t} \right\}\exp \left( {s + \frac{1}{2}v^2 } \right)N\left( {\frac{{s + v^2 + (r - \lambda m - \frac{1}{2}\sigma ^2 )\Delta t}}{v}} \right) \\ & & - N\left( {\frac{{s + (r - \lambda m - \frac{1}{2}\sigma ^2 )\Delta t}}{v}} \right) \\ &=& \left( {1 + m} \right)^n {\rm e}^{\left( {r - \lambda m} \right)\Delta t} N\left( {l_1 } \right) - N\left( {l_2 } \right), \end{eqnarray*} 其中 \[l_1 = \frac{{n\ln \left( {1 + m} \right) - \frac{1}{2}n\theta ^2 + \left( {r - \lambda m - \frac{1}{2}\sigma ^2 } \right)\Delta t}}{{\sqrt {\sigma ^2 \Delta t + n\theta ^2 } }},\qquad {\rm{ }}l_2 = l_1 - \sqrt {\sigma ^2 \Delta t + n\theta ^2 } , \] 所以

$$ {\sigma'} ^2 = \sigma ^2 - \frac{2}{{\Delta t}}\kappa \sum\limits_{n = 0}^\infty {\frac{{{\rm e}^{ - \lambda \Delta t} \left( {\lambda \Delta t} \right)^n }}{{n!}}\left\{ {\left( {1 + m} \right)^n {\rm e}^{\left( {r - \lambda m} \right)\Delta t} \left[{1 - 2N\left( {l_1 } \right)} \right] - \left[{1 - 2N\left( {l_2 } \right)} \right]} \right\}}. (3.6)$$

(3.6)

根据引理3.1,将修正波动率${\sigma'} ^2$ 取代不带有交 易费用的期权定价公式(3.2)中的$\sigma$ 即可得到结论, 证毕.

假设市场股票价格$S$ ,市场利率$r$ ,波动率$\sigma $ ,跳跃强度$\lambda $ ,交易费用 $\kappa $ ,跳跃幅度$m$为梯形模糊数,分别记为 $\mathop S\limits^ \sim ,\mathop r\limits^ \sim ,\mathop \sigma \limits^ \sim ,\mathop \lambda \limits^ \sim , \mathop \kappa \limits^ \sim ,\mathop m\limits^ \sim$.

在得出我们的结论前,先不加证明的给出以下两个引理.

引理4.1     (扩张原理^[14]) 设映射 $f:U_1 \times U_2 \times \cdots \times U_m \to V_1 \times V_2 \times \cdots V_n$ , \[(u_1 ,u_2 ,\cdots ,u_m ) \mapsto f(u_1 ,u_2 ,\cdots ,u_m ) = v = (v_1 ,v_2 ,\cdots ,v_n ), \] 由$f$ 可导出映射 \[\mathop F\limits^ \sim :\psi (U_1 ) \times \psi (U_2 ) \times \cdots \times \psi (U_m ) \to \psi (V_1 \times V_2 \times \cdots V_n ), \] 它具有隶属函数 $$ \mathop F\limits^ \sim (\mathop {A_1 }\limits^ \sim \times \mathop {A_2 }\limits^ \sim \times \cdots \times \mathop {A_m }\limits^ \sim ) = \left\{ \begin{array}{ll} \mathop \vee \limits_{f(u_1 ,u_2 ,\cdots ,u_m ) = v}(\mathop \wedge \limits_{i = 1}^m \mathop A\limits^ \sim _i (u_i )) & v \in f(\prod\limits_{i = 1}^m {A_i } ) ,\\ [3mm] {\rm{0}} & v \notin f(\prod\limits_{i = 1}^m {A_i } ), \end{array} \right. $$ 其中 $v = (v_1 ,v_2 ,\cdots ,v_n ) \in V_1 \times V_2 \times \cdots \times V_n , \mathop {A_i }\limits^ \sim \in \psi (V_i ){\rm{ ,}}A_i \in \varphi (V_i ), {\rm{ }}i = 1,{\rm{ }}2,\cdots ,m$.

引理4.2     (Wu^[6]) 令 $f(x_1 ,\cdots ,x_n )$ 为定义于${\Bbb R}^n$上的连续实值函数,$\mathop {a_1 }\limits^ \sim , \cdots ,\mathop {a_n}\limits^ \sim $ 为$n$个模糊数.令 $\mathop f\limits^ \sim :F^n \to F$ 是一 个由$f(x_1 ,\cdots ,x_n )$ 通过扩展原理引导生产的模糊值函数. 假设对于任意属于$f$ 定义域里面的值$r$, $\{ (x_1 ,\cdots ,x_n ):r = f(x_1 ,\cdots ,x_n )\}$是${\Bbb R}^n$ 中的一个紧子集,那么$\mathop f\limits^ \sim (\mathop {a_1 }\limits^ \sim ,\cdots ,\mathop {a_n }\limits^ \sim )$ 是一个模糊数,且它的$\alpha$ -截集是 \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \Big( {\mathop f\limits^ \sim (\mathop {a_1 }\limits^ \sim , \cdots ,\mathop {a_n }\limits^ \sim )} \Big)_\alpha &=& \{ f(x_1 ,\cdots ,x_n ):x_1 \in (\mathop {a_1 }\limits^ \sim )_\alpha ,\cdots ,(\mathop {a_n }\limits^ \sim )_\alpha \} \\ &=& \{ f(x_1 ,\cdots ,x_n ):(\mathop {a_1 }\limits^ \sim )_\alpha ^L \le x_1 \le (\mathop {a_1 }\limits^ \sim )_\alpha ^U f,\cdots ,(\mathop {a_n }\limits^ \sim )_\alpha ^L \le x_n \le (\mathop {a_n }\limits^ \sim )_\alpha ^U \} \end{eqnarray*}

定理4.1     在模糊环境中,跳扩散模型(2.1)下带交易费 用的欧式看涨期权价格$\mathop C\limits^ \sim $为一个模糊数, 其$\alpha $ -截集为 ${\mathop C\limits^ \sim} _\alpha = \left[{{\mathop C\limits^ \sim}_\alpha ^L ,{\mathop C\limits^ \sim}_\alpha ^U } \right]$,其中 \begin{eqnarray*} % \nonumber to remove numbering (before each equation) {\mathop C\limits^ \sim} _\alpha ^L &=& \sum\limits_{n = 0}^\infty {\frac{{\left[{{\mathop \lambda \limits^ \sim} _\alpha ^L \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)\left( {T - t} \right)} \right]^n }}{{n!}}} \left( {{{\mathop S\limits^ \sim} _t} } \right)_\alpha ^L {\rm e}^{ -{ \mathop \lambda \limits^ \sim } _\alpha ^U \left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right)\left( {T - t} \right)} N\left( {\left( {\mathop {d_1 }\limits^ \sim } \right)_\alpha ^L } \right) \\ & & - \sum\limits_{n = 0}^\infty {\frac{{\left[{{\mathop \lambda \limits^ \sim} _\alpha ^U \left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right)\left( {T - t} \right)} \right]^n }}{{n!}}} {\rm e}^{ - {\mathop r\limits^ \sim} _\alpha ^L \left( {T - t} \right)} KN\left( {\left( {\mathop {d_2 }\limits^ \sim } \right)_\alpha ^U } \right) , \end{eqnarray*} \begin{eqnarray*} {\mathop C\limits^ \sim}_\alpha ^U &=& \sum\limits_{n = 0}^\infty {\frac{{\left[{{\mathop \lambda \limits^ \sim} _\alpha ^U \left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right)\left( {T - t} \right)} \right]^n }}{{n!}}} \left( {{\mathop S\limits^ \sim} _t } \right)_\alpha ^U {\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^L \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)\left( {T - t} \right)} N\left( {\left( {\mathop {d_1 }\limits^ \sim } \right)_\alpha ^L } \right) \\ & & - \sum\limits_{n = 0}^\infty {\frac{{\left[{{\mathop m\limits^ \sim}_\alpha ^L \left( {1 + {\mathop m\limits^ \sim }_\alpha ^L } \right)\left( {T - t} \right)} \right]^n }}{{n!}}} {\rm e}^{ - {\mathop m\limits^ \sim}_\alpha ^U \left( {T - t} \right)} KN\left( {\left( {\mathop {d_2 }\limits^ \sim } \right)_\alpha ^U } \right) , \end{eqnarray*} 其中 $$ \left( {\mathop {d_1 }\limits^ \sim } \right)_\alpha ^L = \frac{{\ln \frac{{\left({{\mathop S\limits^ \sim} _t } \right)_\alpha ^L \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)^n }}{K} + \left( {{\mathop r\limits^ \sim} _\alpha ^L - \lambda {\mathop m\limits^ \sim} _\alpha ^U + \frac{1}{2}\left( {{\mathop {\sigma'} \limits^ \sim} ^2 } \right)_\alpha ^L } \right)\left( {T - t} \right) + \frac{1}{2}n\theta ^2 }}{{\sqrt {\left( {{\mathop {\sigma'} \limits^ \sim } ^2 } \right)_\alpha ^U \left( {T - t} \right) + n\theta ^2 } }} , $$ $$ \left( {\mathop {d_1 }\limits^ \sim } \right)_\alpha ^U = \frac{{\ln \frac{{({\mathop S\limits^ \sim} _t )_\alpha ^U (1 + (\mathop m\limits^ \sim )_\alpha ^U )^n }}{K} + \left( {{\mathop r\limits^ \sim} _\alpha ^U - \lambda {\mathop m\limits^ \sim} _\alpha ^L + \frac{1}{2}\left( {{\mathop {\sigma'}\limits^ \sim} ^2} \right)_\alpha ^U } \right)\left( {T - t} \right) + \frac{1}{2}n\theta ^2 }}{{\sqrt {\left( {{\mathop {\sigma'}\limits^ \sim} ^2 } \right)_\alpha ^L \left( {T - t} \right) + n\theta ^2 } }} , $$ $$ \left( {\mathop {d_2 }\limits^ \sim } \right)_\alpha ^L = \left( {\mathop {d_1 }\limits^ \sim } \right)_\alpha ^L - \sqrt {\left( {{\mathop {\sigma'}\limits^ \sim } ^2 } \right)_\alpha ^U \left( {T - t} \right) + n\theta ^2 } , $$ $$ \left( {\mathop {d_2 }\limits^ \sim } \right)_\alpha ^U = \left( {\mathop {d_1 }\limits^ \sim } \right)_\alpha ^U - \sqrt {\left( {{\mathop {\sigma' }\limits^ \sim} ^2 } \right)_\alpha ^L \left( {T - t} \right) + n\theta ^2 } $$ 且 \begin{eqnarray*} \left( {\mathop {\sigma'}\limits^ \sim} ^2 \right)_\alpha ^L &=& \left( {{\mathop \sigma \limits^ \sim} _\alpha ^L } \right)^2 - \frac{2}{{\Delta t}}{\mathop \kappa \limits^ \sim} _\alpha ^U \sum\limits_{n = 0}^\infty \frac{{{\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^L \Delta t} \left( {{\mathop \lambda \limits^ \sim} _\alpha ^U \Delta t} \right)^n }}{{n!}}\left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right)^n {\rm e}^{\left( {{\mathop r\limits^ \sim} _\alpha ^U - {\mathop \lambda \limits^ \sim} _\alpha ^L {\mathop m\limits^ \sim} _\alpha ^L } \right)\Delta t} \\ &&\times \left[{1 - 2N\left( {\left( {\mathop {l_1 }\limits^ \sim } \right)_\alpha ^L } \right)} \right] + \frac{2}{{\Delta t}}{\mathop \kappa \limits^ \sim} _\alpha ^U \sum\limits_{n = 0}^\infty {\frac{{{\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^L \Delta t} \left( {{\mathop \lambda \limits^ \sim} _\alpha ^U \Delta t} \right)^n }}{{n!}} \left[{1 - 2N\left( {\left( {\mathop {l_2 }\limits^ \sim } \right)_\alpha ^U } \right)} \right]} , \end{eqnarray*} \begin{eqnarray*} \left( {{\mathop {\sigma' }\limits^ \sim} ^2 } \right)_\alpha ^U &=& \left( {{\mathop \sigma \limits^ \sim} _\alpha ^U } \right)^2 - \frac{2}{{\Delta t}}{\mathop \kappa \limits^ \sim} _\alpha ^L \sum\limits_{n = 0}^\infty \frac{{{\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^U \Delta t} \left( {{\mathop \lambda \limits^ \sim} _\alpha ^L \Delta t} \right)^n }}{{n!}} \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)^n {\rm e}^{\left( {{\mathop r\limits^ \sim} _\alpha ^L - {\mathop \lambda \limits^ \sim} _\alpha ^U {\mathop m\limits^ \sim} _\alpha ^U } \right)\Delta t} \\ &&\times \left[{1 - 2N\left( {\left( {\mathop {l_1 }\limits^ \sim } \right)_\alpha ^U } \right)} \right] + \frac{2}{{\Delta t}}{\mathop \kappa \limits^ \sim} _\alpha ^L \sum\limits_{n = 0}^\infty \frac{{{\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^U \Delta t} \left( {{\mathop \lambda \limits^ \sim} _\alpha ^L \Delta t} \right)^n }}{{n!}} \\ && \times \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)^n \left[{1 - 2N\left( {\left( {\mathop {l_2 }\limits^ \sim } \right)_\alpha ^L } \right)} \right] \end{eqnarray*} 且 $$ \left( {\mathop {l_1 }\limits^ \sim } \right)_\alpha ^L = \frac{{\ln \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)^n + \left( {{\mathop r\limits^ \sim} _\alpha ^L - {\mathop \lambda \limits^ \sim} _\alpha ^U {\mathop m\limits^ \sim} _\alpha ^U + \frac{1}{2}\left( {{\mathop \sigma \limits^ \sim} _\alpha ^L } \right)^2 } \right)\Delta t}}{{\sqrt {\left( {{\mathop \sigma \limits^ \sim} _\alpha ^U } \right)^2 \Delta t + n\theta ^2 } }} , $$ $$ \left( {\mathop {l_1 }\limits^ \sim } \right)_\alpha ^U = \frac{{\ln \left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right)^n + \left( {{\mathop r\limits^ \sim} _\alpha ^U - {\mathop \lambda \limits^ \sim} _\alpha ^L {\mathop m\limits^ \sim} _\alpha ^L + \frac{1}{2}\left( {{\mathop \sigma \limits^ \sim} _\alpha ^U } \right)^2 } \right)\Delta t}}{{\sqrt {\left( {{\mathop \sigma \limits^ \sim} _\alpha ^L } \right)^2 \Delta t + n\theta ^2 } }} , $$ $$ \left( {\mathop {l_2 }\limits^ \sim } \right)_\alpha ^L = \left( {\mathop {l_1 }\limits^ \sim } \right)_\alpha ^L - \sqrt {\left( {{\mathop \sigma \limits^ \sim} _\alpha ^U } \right)^2 \Delta t + n\theta ^2 } , $$ $$ \left( {\mathop {l_2 }\limits^ \sim } \right)_\alpha ^U = \left( {\mathop {l_1 }\limits^ \sim } \right)_\alpha ^U - \sqrt {\left( {{\mathop \sigma \limits^ \sim} _\alpha ^L } \right)^2 \Delta t + n\theta ^2 }. $$

证     当$\mathop S\limits^ \sim ,\mathop r\limits^ \sim ,\mathop \sigma \limits^ \sim ,\mathop \lambda \limits^ \sim , \mathop \kappa \limits^ \sim ,\mathop m\limits^ \sim$ 为模糊数时, 根据引理4.2,跳扩散模型(2.1)下的欧式看涨期权价格 $\mathop C\limits^ \sim $ 为一个模糊数.又由于确定的数可以看作其隶属函数为示性函数的广义模糊数, 因而根据引理4.1,期权定价公式(3.4)的模糊化形式如下

\begin{eqnarray} \mathop C\limits^ \sim &= &\sum\limits_{n = 0}^\infty {\frac{{\left[{\mathop \lambda \limits^ \sim \otimes \left( {1_{\left\{ 1 \right\}} \oplus \mathop m\limits^ \sim } \right) \otimes 1_{\left\{ {T - t} \right\}} } \right]^n }}{{1_{\left\{ {n!} \right\}} }}} \left( {{\mathop S\limits^ \sim} _t } \right) \otimes {\rm e}^{ - \mathop \lambda \limits^ \sim \otimes \left( {1_{\left\{ 1 \right\}} \oplus \mathop m\limits^ \sim } \right) \otimes 1_{\left\{ {T - t} \right\}} } \otimes N\left( {\mathop {d_1 }\limits^ \sim } \right) \nonumber \\ &&- \sum\limits_{n = 0}^\infty {\frac{{\left[ {\mathop \lambda \limits^ \sim \otimes \left( {1_{\left\{ 1 \right\}} \oplus \mathop m\limits^ \sim } \right) \otimes 1_{\left\{ {T - t} \right\}} } \right]^n }} {{1_{\left\{ {n!} \right\}} }}} {\rm e}^{ - \mathop r\limits^ \sim \otimes 1_{\left\{ {T - t} \right\}} } \otimes 1_{\left\{ K \right\}} \otimes N\left( {\mathop {d_2 }\limits^ \sim } \right) , \end{eqnarray}

(4.1)

其中 $$ \mathop {d_1 }\limits^ \sim = \frac{{\ln \frac{{\left( {{\mathop S\limits^ \sim} _t } \right) \otimes \left( {1_{\left\{ 1 \right\}} \oplus \mathop m\limits^ \sim } \right)^n }}{{1_{\left\{ K \right\}} }} \oplus \left( {\mathop r\limits^ \sim - \mathop \lambda \limits^ \sim \otimes \mathop m\limits^ \sim \oplus \frac{1}{2}\left( {{\mathop {\sigma' }\limits^ \sim} ^2 } \right)} \right) \otimes 1_{\left\{ {T - t} \right\}} \oplus 1_{\left\{ {\frac{1}{2}n\theta ^2 } \right\}} }}{{\sqrt {\left( {{\mathop {\sigma' }\limits^ \sim} ^2 } \right) \otimes 1_{\left\{ {T - t} \right\}} \oplus 1_{\left\{ {n\theta ^2 } \right\}} } }}, $$ $$ \mathop {d_2 }\limits^ \sim = \mathop {d_1 }\limits^ \sim - \sqrt {{\mathop {\sigma' }\limits^ \sim} ^2 \otimes 1_{\left\{ {T - t} \right\}} \oplus 1_{\left\{ {n\theta ^2 } \right\}} }, $$ \begin{eqnarray*} {\mathop {\sigma' }\limits^ \sim} ^2 &=& {\mathop \sigma \limits^ \sim} ^2 - 1_{\left\{ {\frac{2}{{\Delta t}}} \right\}} \otimes \mathop \kappa \limits^ \sim \otimes \sum\limits_{n = 0}^\infty \frac{{{\rm e}^{ - \mathop \lambda \limits^ \sim \otimes 1_{\left\{ {\Delta t} \right\}} } \otimes \left( {\mathop \lambda \limits^ \sim \otimes 1_{\left\{ {\Delta t} \right\}} } \right)^n }} {{1_{\left\{ {n!} \right\}} }} \otimes \left( {1_{\left \{ 1 \right\}} \oplus \mathop m\limits^ \sim } \right)^n \\ && \otimes {\rm e}^{\left( {\mathop r\limits^ \sim - \mathop \lambda \limits^ \sim \otimes \mathop m\limits^ \sim } \right) \times\otimes 1_{\left\{ {\Delta t} \right\}} } \otimes \left[{1_{\left\{ 1 \right\}} - 1_{\left\{ 2 \right\}} \otimes N\left( {\mathop {l_1 }\limits^ \sim } \right)} \right] \\ &&+ 1_{\left\{ {\frac{2}{{\Delta t}}} \right\}} \otimes \mathop \kappa \limits^ \sim \otimes \sum\limits_{n = 0}^\infty {\frac{{{\rm e}^{ - \mathop \lambda \limits^ \sim \otimes 1_{\left\{ {\Delta t} \right\}} } \otimes \left( {\mathop \lambda \limits^ \sim \otimes 1_{\left\{ {\Delta t} \right\}} } \right)^n }}{{1_{\left\{ {n!} \right\}} }} \otimes \left[{1_{\left\{ 1 \right\}} - 1_{\left\{ 2 \right\}} \otimes N\left( {\mathop {l_2 } \limits^ \sim } \right)} \right]} , \end{eqnarray*} $$ \mathop {l_1 }\limits^ \sim = \frac{{\ln \left( {1 \oplus \mathop m\limits^ \sim } \right)^n \oplus \left( {\mathop r\limits^ \sim - \mathop \lambda \limits^ \sim \otimes \mathop m\limits^ \sim \oplus \frac{1}{2}\left( {\mathop \sigma \limits^ \sim } \right)^2 } \right) \otimes 1_{\left\{ {\Delta t} \right\}} }} {{\sqrt {\left( {\mathop \sigma \limits^ \sim } \right)^2 \otimes 1_{\left\{ {\Delta t} \right\}} \oplus 1_{\left\{ {n\theta ^2 } \right\}} } }}, \mathop {l_2 }\limits^ \sim = \mathop {l_1 }\limits^ \sim - \sqrt {\left( {\mathop \sigma \limits^ \sim } \right)^2 \otimes 1_{\left\{ {\Delta t} \right\}} \oplus 1_{\left\{ {n\theta ^2 } \right\}} }. $$

根据引理4.2,模糊期权价格$\mathop C\limits^ \sim $ 的$\alpha $ -水平集为各个模糊变量的$\alpha $ -水平集在(4.1) 式中运算得到的区间. 由于累积函数 $N\left( x \right)$ 为递增函数,所以 \[ {\mathop {N\left( {\mathop d\limits^ \sim } \right)}\limits^ \sim} _\alpha = \left\{ {N\left( x \right):x \in {\mathop d\limits^ \sim} _\alpha } \right\} = \left\{ {N\left( x \right):{\mathop d\limits^ \sim} _\alpha ^L \le x \le {\mathop d\limits^ \sim} _\alpha ^U } \right\} = \left[{N\left( {{\mathop d\limits^ \sim} _\alpha ^L } \right),N\left( {{\mathop d\limits^ \sim} _\alpha ^U } \right)} \right]. \] 同理,由于指数函数${\rm e}^{ - x} $ 为递减函数,所以 \begin{eqnarray*} \left( {{\rm e}^{ - {\mathop \lambda \limits^ \sim} \otimes \left( {1_{\left\{ 1 \right\}} \oplus \mathop m\limits^ \sim } \right) \otimes 1_{\left\{ {T - t} \right\}} } } \right)_\alpha &=& \left[{{\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^U \left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right) (T - t)} ,{\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^L \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)(T - t)} } \right] , \\ \left( {{\rm e}^{ - \mathop r\limits^ \sim \otimes 1_{\left\{ {T - t} \right\}} } } \right)_\alpha &=& \left[{{\rm e}^{ - {\mathop r\limits^ \sim} _\alpha ^U \left( {T - t} \right)} ,{\rm e}^{ - {\mathop r\limits^ \sim} _\alpha ^{_L } \left( {T - t} \right)} } \right]. \end{eqnarray*} 对数函数$\ln x$为递增函数,所以 \begin{eqnarray*} \left( {\ln \frac{{\left( {{\mathop S\limits^ \sim}_t } \right) \otimes \left( {1_{\left\{ 1 \right\}} \oplus \mathop m\limits^ \sim } \right)^n }}{{1_{\left\{ {\rm{K}} \right\}} }}} \right)_\alpha &=& \left[{\ln \frac{{{{\mathop S\limits^ \sim} _t} _\alpha ^{\rm{L}} }}{K} + n\left( {1{\rm{ + m}}} \right), \ln \frac{{{{\mathop S\limits^ \sim} _t} _\alpha ^{\rm{U}} }}{K} + n\left( {1{\rm{ + m}}} \right)} \right]. \end{eqnarray*}

所以,$\mathop C\limits^ \sim $ 的左右端点分别为 \begin{eqnarray*} {\mathop C\limits^ \sim} _\alpha ^L &=& \sum\limits_{n = 0}^\infty {\frac{{\left[{{\mathop \lambda \limits^ \sim} _\alpha ^L \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)\left( {T - t} \right)} \right]^n }}{{n!}}} \left( {{\mathop S\limits^ \sim} _t } \right)_\alpha ^L {\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^U \left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right)\left( {T - t} \right)} N\left( {\left( {\mathop {d_1 }\limits^ \sim } \right)_\alpha ^L } \right) \\ & & - \sum\limits_{n = 0}^\infty {\frac{{\left[{{\mathop \lambda \limits^ \sim} _\alpha ^U \left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right)\left( {T - t} \right)} \right]^n }}{{n!}}} {\rm e}^{ - {\mathop r\limits^ \sim} _\alpha ^L \left( {T - t} \right)} KN\left( {\left( {\mathop {d_2 }\limits^ \sim } \right)_\alpha ^U } \right) , \\ {\mathop C\limits^ \sim} _\alpha ^U &=& \sum\limits_{n = 0}^\infty {\frac{{\left[{{\mathop \lambda \limits^ \sim} _\alpha ^U \left( {1 + {\mathop m\limits^ \sim} _\alpha ^U } \right)\left( {T - t} \right)} \right]^n }}{{n!}}} \left( {{\mathop S\limits^ \sim} _t } \right)_\alpha ^U {\rm e}^{ - {\mathop \lambda \limits^ \sim} _\alpha ^L \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)\left( {T - t} \right)} N\left( {\left( {\mathop {d_1 }\limits^ \sim } \right)_\alpha ^L } \right) \\ & & - \sum\limits_{n = 0}^\infty {\frac{{\left[{{\mathop \lambda \limits^ \sim} _\alpha ^L \left( {1 + {\mathop m\limits^ \sim} _\alpha ^L } \right)\left( {T - t} \right)} \right]^n }}{{n!}}} {\rm e}^{ - {\mathop r\limits^ \sim} _\alpha ^U \left( {T - t} \right)} KN\left( {\left( {\mathop {d_2 }\limits^ \sim } \right)_\alpha ^U } \right). \end{eqnarray*} 证毕.

由于一个模糊数是由其隶属函数确定的,我们可以得到 $\mathop C\limits^ \sim $ 的隶属函数为^[16]

$$ \mu _{\mathop C\limits^ \sim } \left( c \right) = \mathop {\sup }\limits_{\alpha \in [0,1]} \alpha \cdot 1_{{\mathop C\limits^ \sim} _\alpha } \left( c \right), $$

(4.2)

即期权价格为 $c$ 的隶属度是所有包含 $c$ 的$\alpha $ -截集中 $\alpha $ 的上确界.

由于不同的投资者对决定期权价格的参数的模糊性有不同的度量, 因此,即使在相同的水平$\alpha $下,不同的投资者得到的价 格区间也可能会不同, 这不利于统一定价,因此很有必要将区间数转化成确切数, 该转化过程被称为``退模糊化".本文选取能包含最多信息且具有一 般性的一种退模糊化方法-权 重模糊期望^[17],${\mathop a\limits^ \sim}$ 的权重模糊期望为

$$ M\left( {\mathop a\limits^ \sim } \right) = \int_0^1 {\frac{{f(\alpha )}}{2}\left( {{\mathop a\limits^ \sim} _\alpha ^U + {\mathop a\limits^ \sim} _\alpha ^L } \right){\rm d}\alpha }, (4.3) $$

(4.3)

其中,$f(\alpha )$为权重函数满足 $ \int_0^1 {f(\alpha ){\rm d}\alpha } = 1 . $

考虑一个到期日为一年,敲定价格为90的欧式看涨期权, 此时股票的价格大约为100,波动率大约为26\%,利率大约为5\%, 单位时间内的跳跃次数大约为15,跳跃幅度大约为-0.22\%, 交易费用大约为0.2\%,假设t=0.这些决定期权价格并带有模 糊性的参数我们可以如下的模糊数来描述 $$ \mathop S\limits^ \sim = \left[{98.903,98.907,0.103,0.293} \right],\qquad \mathop \sigma \limits^ \sim = \left[{0.259,0.2603,0.018,0.017} \right] , $$ $$ \mathop r\limits^ \sim = \left[{0.0498,0.0503,0.0018,0.0092} \right],\qquad \mathop \lambda \limits^ \sim = \left[{15,15,0.01,0.02} \right] , $$ $$ \mathop h\limits^ \sim = \left[{ - 0.022,- 0.022,0.004,0.003} \right],\qquad \mathop \kappa \limits^ \sim = \left[{0.00198,0.00202,0.00002,0.00003} \right]. $$

在计算期权价格${\mathop C\limits^ \sim} _\alpha ^L ,$ ${\mathop C\limits^ \sim} _\alpha ^U $的公式中包括两个无穷级数, 修正波动率级数和期权价格表达式里面的级数. 在我们的计算中很容易验证第一个级数取前10项和来逼近修正波动率, 修正波动率的误差已经小于$10^{ - 16} $; 第二个级数取前50项和来逼近期权价格时,期权价格的误差已经小 于$10^{ - 13} $.因此,我们在数值计算中分别用前10项和与前 50项和来逼近两个级数(注: 以下计算均是在Sage软件中完成的).

表 1中,我们给出期权模糊价格的 水平闭区间.比如说对应于 期权模糊价格的$\alpha$ -水平闭区间为[22.68,25.83], 从投资者的角度来考虑,如果投资者对置信水平0.95满意的话, 那么投资者可以选取区间[22.68,25.83]内的任意一个价格值 作为投资时的参考价值.

表 2给出该欧式看涨期权的部分置信水平,图 1给出期权价格的隶属函数. 由图 1我们可以看出该隶属函数为L-R模糊函数,函数在左半边区间是单增的, 在右半区间是单减的,单增部分反映了卖者的满意度随着价格的上升而增大, 单减部分反映了买者的满意度随着价格的上升而减小.当欧式看涨期权的价格 为24.8时,它的置信水平为0.984.因此如果投资者对98.4\%的满意度满意的话, 他可以用24.8作为他购买期权的参考价格.当欧式期权的价格为24.2时, 他的满意度为1.

为了统一定价,我们计算出模糊期权期望价格,我们用梯形模糊数来表 示模糊变量$\mathop S\limits^ \sim ,$ $\mathop r\limits^ \sim ,$ $\mathop \sigma \limits^ \sim ,\mathop \lambda \limits^ \sim , \mathop \kappa \limits^ \sim ,\mathop m\limits^ \sim ,$ 特别地, 选取模糊股票价格、模糊波动率、模糊利率、模糊跳跃强度、 模糊跳跃幅度、模糊交易费用率以及 模糊幅度分别为$\beta _s = 0.103$,$\gamma _s = 0.293$, $\beta _\sigma = 0.018$,$\gamma _\sigma = 0.017$, $\beta _r = 0.0018$, $\gamma _r = 0.0092$,$\beta _\lambda = 0.01$, $\gamma _\lambda = 0.02$,$\beta _m = 0.004$, $\gamma _m = 0.003$, $\beta _\kappa = 0.00002,$ $\gamma _\kappa = 0.00003$.到期日为$T = 1$,取 $f\left( \alpha \right) = 2\alpha $. 表 3中每个参数对应的两个数值,分别代表梯形模糊数的第一个峰值 和第二个峰值.

为了作比较,我们不仅算出了本文模型(CWPM-J)下的期权价格, 还给出了Merton跳扩散模型(MJD)下的期权价格,在表三中, 可以看出由CWPM-J 得到的期权价格比MJD模型得到的期权价格更高一些,这似 乎和我们的直觉理解是一致的,因为模糊环境中的跳扩散模型比MJD模 型包含更多的不确定性, 所以价格应该更高一些.还可以发现,期权的敲定价格越高, 其期权价格越低,这也是与实际一致的,因为对于欧式看涨期权而言, 期权价格是看涨的, 所以敲定价格越高,期权的收益越低,因此期权的价格越低.

若令交易费用外的其它参数不变,考虑交易费用对期权价格的影响, 如表 4.这里选取模糊交易费用的左右扩散幅度分别为0.002\%和0.003\%, 模糊股价、模糊波动率、模糊利率、模糊跳跃幅度、模糊跳跃强度分别为 $$ \mathop S\limits^ \sim = \left[{98.903,98.907,0.103,0.293} \right] ,\qquad \mathop \sigma \limits^ \sim = \left[{0.259,0.2603,0.018,0.017} \right] , $$ $$ \mathop r\limits^ \sim = \left[{0.0543,0.0554,0.0018,0.0092} \right] ,\qquad \mathop h\limits^ \sim = \left[{0.022,0.022,0.004,0.003} \right] , $$ $$ \mathop \lambda \limits^ \sim = \left[{10,10,0.01,0.02} \right] . $$

从表 4中可以看出来在其它参数不变的情况下,期权价格模 糊期望随着交易费用率的增大而增大, 这从直观上理解也是合理的,当交易费用率变大的时候,期权金 应该更加昂贵一些.

本文利用概率方法推导出加入交易费用之后期权定价公式中修正波动率 的具体表达形式, 从而得出跳扩散模型下带有交易费用的欧式期权定价公式.然后, 给出模糊环境下的期权定价方法, 得到模糊价格区间,隶属函数,模糊期望.最后, 利用数学软件进行实证分析得出本文定价方法的合理性与实用性.