0 引言

随着金融市场的不断发展, 期权种类不断增多, 近年来市场上出现了许多新型期权.篮子期权就是新型期权的一种, 由于其在价格上的优势与其本身所具有的灵活性, 使得人们对这种投资组合型期权的需求在不断增加.篮子期权是一种多资产期权, 其收益由多个标的资产的加权平均价格决定, 欧式篮子期权的加权平均价格可分为几何平均和算数平均.对于篮子期权的定价问题的研究始于1973年Merton于文献[1]中在标的股票价格满足Black-Scholes模型下给出的欧式看涨期权定价公式.文献[2-3]给出了其他类型的篮子期权价格公式.文献[4]利用风险中性方法给出了布朗运动环境下欧式几何篮子期权定价公式.文献[5]利用偏微分方程得到了布朗运动环境下欧式几何篮子期权定价公式, 由于几何布朗运动本身的局限性, 使其只能描述股价与过去无关的股票价格, 而无法准确描述股价未来变化情况与过去有关的股票价格.分数布朗运动是一种具有平稳增量但不具有独立增量的高斯过程, 这使基能够描述未来股价与过去有关的股价变动过程.文献[6]利用保险精算法讨论了分数布朗运动环境下欧式篮子期权定价公式.有关分数布朗运动环境下的期权定价问题可参见文献[7-8].作为是分数布朗运动的推广, 双分数布朗运动是一种比分数布朗运动更一般的自相似高斯过程, 它不仅具有分数布朗运动所具有的性质, 且其增量不具有平稳性, 这使其能够描述一些分数布朗运动所不能描述的股价变化过程.文献[9]首次提出了双分数布朗运动的概念.用双分数布朗运动驱动的随机微分方程来刻画资产的价格变化更符合实际的需求, 可以描述更广泛的金融现象^[10-13].目前有关期权定价的方法主要有风险中性方法、偏微分方程方法和保险精算方法等, 其中保险精算方法不但适用于无套利完备的金融市场, 适用于有套利不完备的金融市场, 其思想是将期权定价问题转化为公平保费问题.有关保险精算方法的应用可参见文献[14-15].当前国内外对双分数布朗运动环境下的各种期权定价的研究比较少.本文在双分数布朗运动环境下, 建立更贴合市场的金融数学模型, 对篮子期权定价公式进行推广，得到双分数布朗运动环境下欧式几何看涨、看跌篮子期权定价公式，以及两者之间的平价关系.

1 金融市场模型

定义1^[16]  中心高斯过程(B_t^{H, K}, t≥0) 称为双分数布朗运动, 如果均值为零, 协方差为

$ {R^{H,K}}\left( {t,s} \right) = E\left[ {\begin{array}{*{20}{c}} {B_t^{H,K}}&{B_s^{H.K}} \end{array}} \right] = \frac{1}{{{2^K}}}\left( {{{\left( {{t^{2H}} + {s^{2H}}} \right)}^K} - {{\left| {t - s} \right|}^{2HK}}} \right),s,t \ge 0. $

其中H∈(0, 1), K∈(0, 1].

当K=1时, 双分数布朗运动就退化为分数布朗运动, 当K=1, H=1/2时, 双分数布朗运动就退化为标准布朗运动.

假设股票价格S_i(t) 满足微分方程

$ {\rm{d}}{S_i}\left( t \right) = {S_i}\left( t \right){\mu _i}{\rm{d}}t + {S_i}\left( t \right)\sum\limits_{j = 1}^m {{\sigma _{ij}}{\rm{d}}B_j^{H,K}\left( t \right)} ,0 \le t \le T. $

(1)

其中{B_j^{H, K} (t), t≥0}(j=1, 2, 3, …, m) 表示概率空间(Ω, F, P) 上m个相互独立的双分数布朗运动.0 < H < 1, 0 < K≤1, 其中S_i(t)(i=1, 2, 3, …, n) 表示第i个标的资产在t时刻的价格.第i个资产收益率μ_i与波动率σ_ij均为常数.

引理1  随机微分方程(1) 的解为

$ {S_i}\left( T \right) = {S_i}\left( 0 \right)\exp \left\{ {{\mu _i}T - \frac{1}{2}\sum\limits_{j = 1}^m {\sigma _{ij}^2{T^{2HK}}} + \sum\limits_{j = 1}^m {{\sigma _{ij}}B_j^{H,K}\left( T \right)} } \right\}. $

证明  由双分数布朗运动多维Itô公式可证明.

定义2^[17]  股票价格{S_i(t), t≥0}在[t, T]上的期望回报率β_i(u), u∈[t, T]定义为

$ \exp \left\{ {\int_t^T {{\beta _i}\left( u \right){\rm{d}}u} } \right\} = \frac{{{\rm{E}}\left[ {{S_i}\left( T \right)} \right]}}{{{S_i}\left( t \right)}}. $

引理2  股票价格{S_i(T), T≥0}在[t, T]上的期望回报率β_i(u), u∈[t, T]为

$ {\beta _i}\left( u \right) = {\mu _i},u \in \left[ {t,T} \right]. $

证明  由引理1可知

$ \frac{{\rm{E}\left[ {{S_i}\left( T \right)} \right]}}{{{S_i}\left( t \right)}} = \frac{{\rm{E}\left[ {\left( {{S_i}\left( t \right)\exp \left\{ {{\mu _i}T - \frac{1}{2}\sum\limits_{j = 1}^m {\sigma _{ij}^2{T^{2HK}}} + \sum\limits_{j = 1}^m {{\sigma _{ij}}B_j^{H,K}\left( T \right)} } \right\}} \right)} \right]}}{{{S_i}\left( t \right)}}. $

又因为

$ \rm{E}\left\{ {\exp \left( {\sum\limits_{j = 1}^m {{\sigma _{ij}}B_j^{H,K}\left( T \right)} } \right)} \right\} = \exp \left\{ {\frac{1}{2}\sum\limits_{j = 1}^m {\sigma _{ij}^2{T^{2HK}}} } \right\}. $

所以

$ \exp \left\{ {\int_t^T {{\beta _i}\left( u \right){\rm{d}}u} } \right\} = \exp \left\{ {{\mu _i}T} \right\}. $

2 篮子期权定价公式

欧式几何篮子期权的收益函数为

$ C = {\left( {\prod\limits_{i = 1}^n {S_i^{\alpha i}\left( T \right) - X} } \right)^ + }. $

其中T为到期日, X为看涨期权的执行价格, α_i是第i种资产S_i在篮子期权中所占的比例, 且$\sum\limits_{i=1}^{n}{{{\alpha }_{i}}=1, {{\alpha }_{i}}\ge 0}$.

定义3  (ⅰ) 欧式几何看涨篮子期权在0时刻的保险精算价格定义为

$ {C_n} = \rm{E}\left[ {\prod\limits_{i = 1}^n {{{\left[ {\exp \left( { - {\beta _i}T} \right){S_i}\left( T \right)} \right]}^{\alpha i}} - \exp \left( { - rT} \right)X} } \right]{I_{\left\{ {\prod\limits_{i = 1}^n {{{\left[ {\exp \left( { - {\beta _i}T} \right){S_i}\left( T \right)} \right]}^{\alpha i}} > \exp \left( { - rT} \right)X} } \right\}}}, $

(2)

(ⅱ) 欧式几何看跌篮子期权在0时刻的保险精算价格定义为

$ {P_n} = \rm{E}\left[ {\exp \left( { - rT} \right)X - \prod\limits_{i = 1}^n {{{\left[ {\exp \left( { - {\beta _i}T} \right){S_i}\left( T \right)} \right]}^{\alpha i}}} } \right]{I_{\left\{ {\exp \left( { - rT} \right)X > \prod\limits_{i = 1}^n {{{\left[ {\exp \left( { - {\beta _i}T} \right){S_i}\left( T \right)} \right]}^{\alpha i}}} } \right\}}}. $

(3)

定理1  欧式几何看涨篮子期权在0时刻的保险精算价格为

$ {C_n} = S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2 - {\delta ^2}} } } \right){T^{2HK}}} \right\}\mathit{\Phi }\left( {{d_1}} \right) - \exp \left\{ { - rT} \right\}X\mathit{\Phi }\left( {{d_2}} \right). $

(4)

其中

$ \begin{array}{*{20}{c}} {{d_1} = \ln \frac{{S\left( 0 \right)}}{X} + rT - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } + {\delta ^2}{T^{2HK}},{d_2} = {d_1} - \delta {T^{HK}},}\\ {{\delta ^2} = \sum\limits_{j = 1}^m {\sum\limits_{l = 1,k = 1}^n {{\alpha _j}{\alpha _k}{\sigma _{lj}}{\sigma _{kj}}} } ,S\left( 0 \right) = \prod\limits_{i = 1}^n {S_i^{\alpha i}\left( 0 \right)} .} \end{array} $

证明  由$\prod\limits_{i=1}^{n}{{{\left[\exp \left(-{{\beta }_{i}}T \right){{S}_{i}}\left( T \right) \right]}^{\alpha i}}>\exp \left( -rT \right)X}$可得

$ S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } + \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}{\sigma _{ij}}B_j^{H,K}\left( T \right)} } } \right\} > \exp \left( { - rT} \right)X. $

令

$ \xi = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}{\sigma _{ij}}B_j^{H,K}\left( T \right)} } , $

(5)

将式(5) 化简后两边取对数得

$ \xi > \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } - rT + \ln \frac{X}{{S\left( 0 \right)}}. $

求ξ的期望与方差.{B_t^{H, K}, t≥0}是均值为0, 协方差为

$ {\mathop{\rm cov}} \left( {B_t^{H,K},B_s^{H,K}} \right) = \frac{1}{{{2^K}}}\left( {{{\left( {{t^{2H}} + {t^{2H}}} \right)}^K} + {{\left| {t - s} \right|}^{2HK}}} \right) $

的高斯过程, 因此可得

$ \begin{array}{l} {\rm{E}}\left( \xi \right) = {\rm{E}}\left[ {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2B_j^{H,K}\left( T \right)} } } \right] = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{\rm{E}}\left[ {B_j^{H,K}\left( T \right)} \right] = 0} } ,\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathop{\rm var}} \left( \xi \right) = {\mathop{\rm var}} \left[ {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2B_j^{H,K}\left( T \right)} } } \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2} } } \right)^2}{\mathop{\rm var}} \left[ {B_j^{H,K}\left( T \right)} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^m {\sum\limits_{l = 1,k = 1}^n {{\alpha _j}{\alpha _k}{\sigma _{lj}}{\sigma _{kj}}{T^{2HK}}} } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\delta ^2}{T^{2HK}}. \end{array} $

令

$ d = \ln \frac{{S\left( 0 \right)}}{X} + tT - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } . $

(6)

由式(6) 可得

$ \begin{array}{l} {\rm{E}}\left[ {\prod\limits_{i = 1}^n {{{\left[ {\exp \left( { - {\beta _i}T} \right){S_i}\left( T \right)} \right]}^{\alpha i}}{I_{\left\{ {\prod\limits_{i = 1}^n {{{\left[ {\exp \left( { - {\beta _i}T} \right){S_i}\left( T \right)} \right]}^{\alpha i}} > \exp \left( { - rT} \right)X} } \right\}}}} } \right] = \\ {\rm{E}}\left[ {S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {\sigma _{ij}^2{T^{2HK}}} } + \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}{\sigma _{ij}}B_j^{H,K}} } } \right\}{I_{\left\{ {\prod\limits_{i = 1}^n {\left[ {\exp \left( { - {\beta _i}T} \right){S_i}\left( T \right)} \right]\alpha i > \exp \left( { - rT} \right)X} } \right\}}}} \right] = \\ \int_{ - d}^{ + \infty } {S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {\sigma _{ij}^2{T^{2HK}}} } + x} \right\}} \frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} \delta {T^{HK}}}}\exp \left( { - \frac{{{x^2}}}{{2{\delta ^2}{T^{2HK}}}}} \right){\rm{d}}x = \\ \frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} \delta {T^{HK}}}}\int_{ - d}^{ + \infty } {S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{t^{2HK}}} } + x - \frac{{{x^2}}}{{2{\delta ^2}{T^{2HK}}}}} \right\}} {\rm{d}}x = \\ \frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} \delta {T^{HK}}}}\int_{ - d}^{ + \infty } {S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } + \frac{{2{\delta ^2}{T^{2HK}} - {x^2}}}{{2{\delta ^2}{T^{2HK}}}}} \right\}} {\rm{d}}x = \\ \frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} \delta {T^{HK}}}}\int_{ - d}^{ + \infty } {S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } + \frac{{ - \left( {x - {\delta ^2}{T^{2HK}}} \right)}}{{2{\delta ^2}{T^{2HK}}}} + \frac{{{\delta ^2}{T^{2HK}}}}{2}} \right\}} {\rm{d}}x = \\ S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } + \frac{{{\delta ^2}{T^{2HK}}}}{2}} \right\}\int_{ - d}^{ + \infty } {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} \delta {T^{HK}}}}} \exp \left( { - \frac{{{{\left( {x - {\delta ^2}{T^{2HK}}} \right)}^2}}}{{2{\delta ^2}{T^{2HK}}}}} \right){\rm{d}}x = \\ S\left( 0 \right)\exp \left\{ { - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } + \frac{{{\delta ^2}{T^{2HK}}}}{2}} \right\}\int_{ - d1}^{ + \infty } {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}} \exp \left( { - \frac{{{y^2}}}{2}} \right){\rm{d}}y = \\ S\left( 0 \right)\exp \left\{ { - \frac{1}{2}{T^{2HK}}\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2 - {\delta ^2}} } } \right)} \right\}\mathit{\Phi }\left( {{d_1}} \right). \end{array} $

且

$ \begin{array}{l} {\rm{E}}\left[ {\exp \left( { - rT} \right)X{I_{\left\{ {\prod\limits_{i = 1}^n {\left[ {\exp \left( { - {\beta _i}T} \right){S_i}\left( T \right)} \right]\alpha i > \exp \left( { - rT} \right)X} } \right\}}}} \right] = \\ \exp \left( { - rT} \right)X\int_{ - d}^{ + \infty } {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} \delta {T^{HK}}}}} \exp \left( { - \frac{{{x^2}}}{{2{\delta ^2}{T^{2HK}}}}} \right){\rm{d}}x = \\ \exp \left( { - rT} \right)X\int_{ - d2}^{ + \infty } {\frac{1}{{\sqrt {2{\rm{ \mathsf{ π} }}} }}} \exp \left( { - \frac{{{y^2}}}{2}} \right){\rm{d}}y = \\ X\exp \left( { - rT} \right)\mathit{\Phi }\left( {{d_2}} \right). \end{array} $

因此得证

$ {C_n} = S\left( 0 \right)\exp \left\{ { - \frac{1}{2}{T^{2HK}}\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2 + {\delta ^2}} } } \right)} \right\}\mathit{\Phi }\left( {{d_1}} \right) - X\exp \left( { - rT} \right)\mathit{\Phi }\left( {{d_2}} \right). $

定理2  欧式几何篮子看跌期权的保险精算价格为

$ {P_n} = X\exp \left( { - rT} \right)\mathit{\Phi }\left( { - {d_2}} \right) - S\left( 0 \right)\exp \left\{ { - \frac{1}{2}{T^{2HK}}\left( {\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2 + {\delta ^2}} } } \right)} \right\}\mathit{\Phi }\left( { - {d_1}} \right). $

其中

$ \begin{array}{*{20}{c}} {{d_1} = \ln \frac{{S\left( 0 \right)}}{X} + rT - \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {{\alpha _i}\sigma _{ij}^2{T^{2HK}}} } + {\delta ^2}{T^{2HK}},{d_2} = {d_1} - \delta {T^{HK}},}\\ {{\delta ^2} = \sum\limits_{j = 1}^m {\sum\limits_{l = 1,k = 1}^n {{\alpha _j}{\alpha _k}{\sigma _{lj}}{\sigma _{kj}}} } ,S\left( 0 \right) = \prod\limits_{i = 1}^n {S_i^{\alpha i}\left( 0 \right)} .} \end{array} $

证明  类似定理1的证明.

推论1  双分数布朗运动下, 欧式几何篮子期权看涨与看跌的平价关系为

$ {P_n} - {C_n} = X\exp \left( { - rT} \right) - {S_i}\left( 0 \right)\exp \left\{ {{\mu _i}T - \frac{1}{2}\sum\limits_{j = 1}^m {\sigma _{ij}^2{T^{2HK}}} + \sum\limits_{j = 1}^m {{\sigma _{ij}}B_j^{H,K}\left( T \right)} } \right\}. $

注  当K=1时, 可得分数布朗运动环境下的欧式几何篮子期权定价公式^[6].特别地当H=1/2时, 可得布朗运动环境下的欧式几何篮子期权定价公式^[5].当n=1时, 可得分数布朗运动环境下欧式期权定价公式^[18].

3 结束语

本文在传统模型的基础上, 采用双分数布朗运动刻画金融市场的资产价格, 使其更符合实际, 利用随机分析理论与保险精算方法, 得到了双分数布朗运动环境下篮子期权定价公式, 并对分数布朗运动环境下的篮子期权定价的相关结论进行了推广, 使其更具有实际意义.本文假定利率为常数, 对于非常数利率情形下的篮子期权定价问题还有待于进一步研究.