0 引言

近年来关于抛物型随机偏微分方程模型的研究越来越受到学者们的关注, 抛物型随机偏微分方程理论已经被广泛应用于随机过程与随机分析、金融数学与金融工程、运筹学与控制论等领域.混合分数布朗运动(mfBm)模型是一类特殊的抛物型随机微分方程, 是高斯过程的衍生过程.众所周知的期权定价理论是金融数学和金融工程的核心理论, 混合分数布朗运动是布朗运动和分数布朗运动的线性组合, 已经被广泛地应用于期权定价理论.文献[1]最早把布朗运动和分数布朗运动组合在一起研究了欧式期权的定价；文献[2]研究了混合分数布朗运动样本轨道的Holder连续性和自相似性; 文献[3-6]已经多次应用跳-扩散模型来刻画股票价格的随机跳行为, 并给出了相应的期权定价公式, 但这种跳-扩散模型无法处理Wick积分, 同时无法定义适合的随机积分来刻画股价的变化, 于是使用混合分数布朗运动来刻画金融资产的波动过程是比较合理的^[7-10].由于分数布朗运动的Itô公式和分数Wick-Itô-Skorohod积分所建立的Black-Scholes (简称B-S模型)已经远远超越了B-S模型的定义和属性, 学者们发现所建立的分数B-S模型不能准确地描述资产的浮动收益和金融市场的波动情形^[9].事实上, 由于分数布朗运动的自相似性、厚尾性和长程关联性, 使得分数布朗运动既不是Markov过程又不是半鞅, 这给随机分析和随机计算带来了极大的困难.于是有些学者^[11-12]提出用混合跳-扩散分数布朗运动(mj-dfBm)模型来刻画金融市场的波动行为.

本文研究一类特殊的混合跳-扩散分数布朗运动模型, 不同于文献[12]的混合跳-扩散分数布朗运动模型, 本文的模型是基于标准布朗运动、分数布朗运动、Poisson过程的线性组合, 利用Itô公式和分数Wick-Itô-Skorohod积分建立了一个新的市场定价模型, 给出了连续支付红利的欧式固定履约和浮动履约回望期权的定价公式, 为了刻画利率的异常波动情形.最后给出的数值模拟与仿真验证了模型的有效性.结果表明, 本文的混合跳-扩散分数布朗运动模型便于计算, 可操作性强, 不但能有效地刻画金融市场的隐含波动率的变化, 而且能够合理地解释金融市场的“微笑现象”.

1 预备及引理

定义1^[1] 设$0 < H < 1$, Hurst参数为$H$的分数布朗运动, 是一个连续高斯过程$\{B_t^H \}_{t\geqslant 0} $, 定义含有参数$\alpha, \beta $和$H$的混合分数布朗运动$M_t^H $, 是由Hurst参数为$H$的分数布朗运动和标准布朗运动的线性组合, 定义概率空间$(\Omega, F, \mathrm {P})$, 对任意的$t\in {\rm {\bf R}}^+$, 满足

$ M_t^H = \alpha {B_t} + \beta B_t^H, $

其中$B_t $是标准布朗运动, $B_t^H $是含有参数$H$的独立分数布朗运动, $\alpha, \beta $是两个给定的实数且满足$(\alpha, \beta )\ne (0, 0)$.

性质1^[2] 混合分数布朗运动$M^H_t$具有如下性质.

(ⅰ)对任意的$H\in (0, 1)\backslash \left\{ {\frac{1}{2}} \right\}$, $M^H_t$是中心高斯过程；

(ⅱ) $M_0^H =0$ (几乎必然);

(ⅲ)对任意的$t, s\in {\rm {\bf R}}^+, M_t^H (\alpha, \beta )$和$M_s^H (a, b)$的协方差函数为

$ {\rm{Cov}}(M_t^H, M_s^H) = {\alpha ^2}(t \wedge s) + \frac{{{\beta ^2}}}{2}({t^{2H}} + {s^{2H}}-|t-s{|^{2H}}), $

其中$\wedge $表示两个数中取最小;

(ⅳ)对任意的$h>0$, $M_{ht}^H (\alpha, \beta )$的独立平稳增量具有自相似性,

$ M_{ht}^H\left( {\alpha, \beta } \right) \buildrel \Delta \over = M_t^H\left( {\alpha {h^{\frac{1}{2}}}, \beta {h^H}} \right), $

其中$\mathop =\limits^\Delta $表示具有相同的属性和规律；

(ⅴ)当$\frac{1}{2} < H < 1$时, $M_{t}^H (\alpha, \beta )$的独立平稳增量是正相关的; 当$H=\frac{1}{2}$时, $M_{t}^H (\alpha, \beta )$的独立平稳增量是不相关的; 当$0 < H < \frac{1}{2}$时, $M_{t}^H (\alpha, \beta )$的独立平稳增量是负相关的；

(ⅵ) $M_{t}^H (\alpha, \beta )$的独立平稳增量具有长程关联性, 当且仅当$H>\frac{1}{2}$；

(ⅶ)对任意的$t\in {\rm {\bf R}}^+$, $M_{t}^H (\alpha, \beta )$满足

$ \rm{E}\left[{{{\left( {M_t^H\left( {\alpha, \beta } \right)} \right)}^n}} \right] = \left\{ \begin{array}{l} 0, \;\;\;\;n = 2l + 1, \\ \frac{{\left( {2l} \right)!}}{{{2^l}l!}}{\left( {{\alpha ^2}t + {\beta ^2}{t^{2H}}} \right)^l}, \;\;n = 2l. \end{array} \right. $

性质的证明详见文献[2].

引理1^[9] 对任意的$0<t<t$, $\sigma \in {\bf c}$, 则

$ {{\tilde E}_t}\left[{{{\rm{e}}^{\sigma \left( {{B_T} + B_T^H} \right)}}} \right] = {{\rm{e}}^{\sigma \left( {{B_T} + B_T^H} \right) + \frac{1}{2}{\sigma ^2}\left( {T -t} \right) + \frac{1}{2}{\sigma ^2}\left( {{T^{2H}} -{t^{2H}}} \right)}}, $

其中$\widetilde {E}_t $是关于风险中性测度的拟条件期望.

引理2^[9] 设二元函数$f(B_T, B_T^H )$满足$\widetilde {E}_t [f(B_T, B_T^H )] < \infty $.则对任意的$0 < t < T$, $\sigma\in {\bf C}$, 有

$ {{\tilde E}_t}\left[{f\left( {\sigma {B_T} + \sigma B_T^H} \right)} \right] = \int_{\bf{R}} {\frac{1}{{\sqrt {2\pi \left[{{\sigma ^2}\left( {T-t + {T^{2H}}-{t^{2H}}} \right)} \right]} }}\exp \left[{-\frac{{{{\left( {x-\sigma {B_t}-\sigma B_t^H} \right)}^2}}}{{2{\sigma ^2}\left( {T - t + {T^{2H}} - {t^{2H}}} \right)}}} \right] \cdot f\left( x \right){\rm{d}}x} . $

引理3^[9] 设$A\in { B}( {\bf R})$, $f(x)=\mathrm {1}_A (x)\mathrm {d}x$, 则

$ {{\tilde E}_t}\left[{{1_A}\left( {\sigma {B_T} + \sigma B_T^H} \right)} \right] = \int_{\bf{R}} {\frac{{\exp }}{{\sqrt {2\pi \left[{{\sigma ^2}\left( {T-t + {T^{2H}}-{t^{2H}}} \right)} \right]} }}\left[{-\frac{{{{\left( {x-\sigma {B_t}-\sigma B_t^H} \right)}^2}}}{{2{\sigma ^2}\left( {T - t + {T^{2H}} - {t^{2H}}} \right)}}} \right] \cdot {1_A}\left( x \right){\rm{d}}x} . $

设$\sigma _1, \sigma _2 \in {\rm {\bf R}}$, 考虑如下随机过程

$ Z_t^ * = {\sigma _1}B_t^ * + {\sigma _2}{\left( {B_t^H} \right)^ * } = {\sigma _1}{B_t} + \sigma _1^2t + {\sigma _2}B_t^H + \sigma _2^2B_t^{2H}, \;\;\;\;\;\;0 \le t \le T, $

其中$\sigma _1, \sigma _2 $是常数并且$(\sigma _1, \sigma _2 )\ne (0, 0)$.根据Girsanov定理, 存在概率测度$P^\ast $, 使得随机过程$Z_t^\ast $是新生成的混合分数布朗运动.定义如下关于概率测度$P^*$的拟条件期望$E_t^\ast [\cdot]$:

$ E_t^ * \left[{{{\rm{e}}^{{\sigma _1}{B_T} + {\sigma _2}{{\left( {B_T^H} \right)}^ * }}}} \right] = {{\rm{e}}^{{\sigma _1}\left( {{B_T} + B_T^H} \right) + \frac{1}{2}\sigma _1^2\left( {T -t} \right) + \frac{1}{2}\sigma _2^2\left( {{T^{2H}} -{t^{2H}}} \right)}}. $

引理4^[9] 设$\widetilde {E}_t [f(\sigma _1 B_T +\sigma _2 B_T^H )] < \infty $, 记$X_t =\mathrm {exp(}-\sigma _1 B_t -\frac{1}{2}\sigma _1^2 t-\sigma _2 B_t^H -\frac{1}{2}\sigma _2^2 t^{2H})$, 对任意的$0 < t < T$, 有

$ {{\tilde E}_t}\left[{f\left( {{\sigma _1}{B_T} + {\sigma _2}B_T^H} \right)} \right] = \frac{1}{{{X_t}}}{{\tilde E}_t}\left[{f\left( {{\sigma _1}{B_T} + {\sigma _2}B_T^H} \right){X_T}} \right]. $

引理5^[9] 定义代数流$F\in L^2$上有界测度$F_T^H $的价格过程$F_t $, $0 < t < T$, 记

$ {X_t} = \exp \left( {-{\sigma _1}{B_t}-\frac{1}{2}\sigma _1^2t-{\sigma _2}B_t^H - \frac{1}{2}\sigma _2^2{t^{2H}}} \right), $

则$F_t$可表示为

$ {F_t} = {{\rm{e}}^{- r\left( {T- t} \right)}}{{\tilde E}_t}\left[F \right], $

其中$r$是常数无风险利率.

引理1-引理5的证明详见文献[1, 9, 10].

2 混合跳扩散分数布朗运动定价模型与分数Wick-Itô-Skorohod积分 2.1 混合跳扩散分数布朗运动定价模型

假设某一支期权的价值$V_t =V(S_t, t)$依赖于$S_t $和$t$, 动态股票价格满足如下随机微分方程

$ {\rm{d}}{S_t} = \mu {\rm{d}}t + {\sigma _1}{\rm{d}}B_t^H + {\sigma _2}{\rm{d}}{B_t}, $

(1)

其中$\mu $是期望收益率, $\sigma _1 $, $\sigma _2 $, $B_t $, $B_t^H $的定义见引理3, $B_t$和$B_t^H$是相互独立的.

引理6^[11] 设$V_t=V(S, t)$是二元可微函数.若随机过程$S_t $适合随机微分方程

$ {\rm{d}}{S_t} = \mu {S_t}{\rm{d}}t + {\sigma _1}{S_t}{\rm{d}}B_t^H + {\sigma _2}{S_t}{\rm{d}}{B_t}, $

则

$ {\rm{d}}{V_t} = \left( {\frac{{\partial V}}{{\partial t}} + \mu {S_t}\frac{{\partial V}}{{\partial S}} + H\sigma _1^2{S_t}{t^{2H-1}}\frac{{{\partial ^2}V}}{{\partial {S^2}}} + \frac{1}{2}\sigma _2^2{S_t}\frac{{{\partial ^2}V}}{{\partial {S^2}}}} \right){\rm{d}}t + {\sigma _1}{S_t}\frac{{\partial V}}{{\partial S}}{\rm{d}}B_t^H + {\sigma _2}{S_t}\frac{{\partial V}}{{\partial S}}{\rm{d}}{B_t}. $

(2)

在完备概率空间$(\Omega, F, F_t, \mathrm {P})$中考虑一个连续时间的金融市场, 其中$\{F_t \}_{0\leqslant t\leqslant T} $是基于布朗运动$B_t$、分数布朗运动$B_t^H$、Poisson过程$P_t $生成的自然$\sigma $-代数流, $P_t$是定义在空间$(F_t, \mathrm {P})$上的Poisson跳过程, 跳跃强度为$\lambda $, 并且$B_t$, $B_t^H$, $P_t$是相互独立的随机过程. $\{j_t \}_{t\geqslant 0} $为复合Poisson过程, 表示为$j_t =\sum\limits_{i=0}^{N_t } {P_i } $.记在Merton假设下, 有$\ln (1+j_t )\sim N[\ln (1+\mu _j )-\frac{1}{2}\sigma _j^2, \sigma _j^2]$, $\mu _j ={\rm E}(P_i )=\exp \{\ln (1+j_t )-1\}$, $\sigma _j^2 =\mathrm {Var}\ln (1+j_t )$, 显然$\mu _j $是关于$j_t $的无条件期望, 是可以计算的一个常数, $j_t$是一个关于时间$t$的有界函数, 所以本文中我们假设$\ln [1+\mu _j]=\sigma _3 $是一个常数.根据引理6, 如下引理成立.

引理7 设$V_t =V(S_t, t)$, $V$是二元可微函数.若随机过程$S_t $适合随机微分方程

$ {\rm{d}}{S_t} = \mu {S_t}{\rm{d}}t + {\sigma _1}{S_t}{\rm{d}}B_t^H + {\sigma _2}{S_t}{\rm{d}}{B_t} + {\sigma _3}{S_t}{\rm{d}}{P_t}, $

(3)

则

$ \begin{array}{l} {\rm{d}}{V_t} = \left[{\frac{{\partial V}}{{\partial t}} + \mu {S_t}\frac{{\partial V}}{{\partial S}} + \left( {H\sigma _1^2{t^{2H-1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right)S_t^2\frac{{{\partial ^2}V}}{{\partial {S^2}}}} \right]{\rm{d}}t + {\sigma _1}{S_t}\frac{{\partial V}}{{\partial S}}{\rm{d}}B_t^H + {\sigma _2}{S_t}\frac{{\partial V}}{{\partial S}}{\rm{d}}{B_t}\\ \;\;\;\;\;\;\;\; + {\sigma _3}{S_t}\frac{{\partial V}}{{\partial S}}{\rm{d}}{P_t} + \lambda {\rm{E}}\left[{V\left( {S\left( {1 + {j_t}} \right), t} \right)-V\left( {S, t} \right)} \right]{\rm{d}}t, \end{array} $

(4)

其中$\sigma _3 S_t \frac{\partial V}{\partial S}\mathrm dP_t $表示Poisson跳过程$\mathrm dt$时间内作用于$\frac{\partial V}{\partial S}$的变化量, $\lambda {\rm E}[V(S(1+j_t ), t)-V(S, t)]\mathrm dt$为Poisson跳跃过程在$t$时间内作用于$V$的变化量, ${\rm E}$表示二元可微函数$V$的期望算子.

证 明 假设$\mathrm dP_t $服从如下形式的两点分布:

$ {\rm{d}}{P_t} = \left\{ \begin{array}{l} {\rm{P}}\left\{ {{j_t} = {\omega _1}} \right\} = p, \;\;\;\;不发生跳, \\ {\rm{P}}\left\{ {{j_t} = {\omega _2}} \right\} = 1-p, \;\;\;发生跳. \end{array} \right. $

在时间区间$[t, t+\mathrm {d}t]$内, 可以确定事件$\omega _1 $发生的概率为P$(\omega _1 )=1-\lambda \mathrm {d}t$, 事件$\omega _2 $发生的概率为P$(\omega _2 )=\lambda \mathrm {d}t$, 记$S_{t+} >S_t $表示股票价格$S_t $在$t$时刻上跳, $S_{t+} < S_t $表示股票价格$S_t $在$t$时刻下跳, 注意到$j_t =\sum\limits_{i=0}^{N_t } {P_i } $非负, 于是$S_{t+} =S_t (1+j_t )>0$.

设$\prod _t =V_t -\nabla _t S_t $是通过简单的对冲原理$\nabla $得到一个金融衍生产品, 选取适当的$\nabla $使得投资组合$\prod $在$(t, t+\mathrm dt)$上是无风险的, 即$\mathrm d\prod =r\prod \mathrm dt$, 根据模型(3), 投资组合$\prod$有如下两种情况.

(ⅰ)如果跳事件不发生, 对于二元可微函数$V_t=V(S, t)$, 利用Itô公式, 有

$ \begin{array}{l} {\rm{d}}\prod\limits_t {\left( {{\omega _1}} \right)} = {\rm{d}}{V_t}- {\nabla _t}{\rm{d}}{S_t}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\left[{\frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H-1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right)S_t^2\frac{{{\partial ^2}V}}{{\partial {S^2}}}} \right]_{\left( {{S_t}, t} \right)}} + {\left( {\frac{{\partial V}}{{\partial S}} -\nabla } \right)_{\left( {{S_t}, t} \right)}}{\rm{d}}{S_t}. \end{array} $

(ⅱ)如果跳事件发生, 则

$ {\rm{d}}\prod\limits_t {\left( {{\omega _2}} \right)} = V\left( {{S_{t + }}} \right)- V\left( {{S_t}, t} \right)- {\nabla _t}\left( {{S_{t + }}- {S_t}} \right) = V\left[{{S_t}\left( {1 + {j_t}} \right), t} \right] -V\left( {{S_t}, t} \right) -{\nabla _t}{j_t}{S_t}. $

于是

$ \begin{array}{l} E\left( {{\rm{d}}\prod\limits_t {} } \right) = r\prod\limits_t {{\rm{d}}t} = \left( {1- \lambda {\rm{d}}t} \right)\left[{{\rm{d}}\prod\limits_t {\left( {{\omega _1}} \right)} } \right] + \lambda {\rm{d}}t\left[{{\rm{d}}\prod\limits_t {\left( {{\omega _2}} \right)} } \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \left( {1 - \lambda {\rm{d}}t} \right)\left\{ {{{\left[{\frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H-1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right)S_t^2\frac{{{\partial ^2}V}}{{\partial {S^2}}}} \right]}_{\left( {{S_t}, t} \right)}} + {{\left( {\frac{{\partial V}}{{\partial S}} - \nabla } \right)}_{\left( {{S_t}, t} \right)}}{\rm{d}}{S_t}} \right\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \lambda {\rm{d}}t\left\{ {V\left[{{S_t}\left( {1 + {j_t}} \right), t} \right] -V\left( {{S_t}, t} \right) -{\nabla _t}{j_t}{S_t}} \right\}. \end{array} $

设$\nabla _t =\frac{\partial V}{\partial S}\left| {_{(S_t, t)} } \right. $, 在上式两边对$j_t$取期望, 并消去$\mathrm dt^2$项, 则

$ \begin{array}{l} \frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H- 1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right){S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}}{\rm{d}}t + \left( {r- \lambda {\sigma _3}} \right)S\frac{{\partial V}}{{\partial S}}\\ \;\;\;\;\;\; + \lambda {\rm{E}}\left[{V\left( {S\left( {1 + {j_t}} \right), t} \right)-V\left( {S, t} \right)} \right] -\left( {r + \lambda } \right)V = 0. \end{array} $

于是在混合跳-扩散分数布朗运动模型中, 欧式期权定价模型可以表示为含有期望算子的抛物型积分方程.

仿照引理6的证明, 对二元可微函数$\mathrm {d}V_t $泰勒展开, 则有

$ {\rm{d}}{V_t} = \frac{{\partial V}}{{\partial t}}{\rm{d}}t + \frac{{\partial V}}{{\partial S}}{\rm{d}}{S_t} + \frac{1}{2} \cdot \frac{{{\partial ^2}V}}{{\partial {S^2}}}{\left( {{\rm{d}}{S_t}} \right)^2} + o\left( {{\rm{d}}t{\rm{d}}{S_t}} \right). $

(5)

由于Cov$(M_t^H, M_s^H)=\alpha^2(t\wedge s)+\frac{\beta^2}{2}(t^{2H}+s^{2H}-|t-s|^{2H})$, 则E$[B_t^H]^2=t^{2H}$, 即可以近似地认为$(\mathrm dB_t)^2=\mathrm dt$, $(\mathrm dB_t^H )^2=2Ht^{2H-1}\mathrm dt$, $(\mathrm dt)^2=0$, 并且$(\mathrm dP_t)^2=\lambda \mathrm dt$, 则

$ \begin{array}{l} {\left( {{\rm{d}}{S_t}} \right)^2} = {\left( {\mu {S_t}{\rm{d}}t + {\sigma _1}{S_t}{\rm{d}}B_t^H + {\sigma _2}{S_t}{\rm{d}}{B_t} + {\sigma _3}{S_t}{\rm{d}}{P_t}} \right)^2}\\ \;\;\;\;\;\;\;\;\;\; = {\mu ^2}S_t^2{\left( {{\rm{d}}t} \right)^2} + \sigma _1^2S_t^2{\left( {{\rm{d}}B_t^H} \right)^2} + \sigma _2^2S_t^2{\left( {{\rm{d}}{B_t}} \right)^2} + \sigma _3^2S_t^2{\left( {{\rm{d}}{P_t}} \right)^2} + 2\left[{\mu {S_t}{\sigma _1}{S_t}{\rm{d}}B_t^H{\rm{d}}t} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\; + \mu {S_t}{\sigma _2}{S_t}{\rm{d}}{B_t}{\rm{d}}t + \mu {S_t}{\sigma _3}{S_t}{\rm{d}}{P_t}{\rm{d}}t + {\sigma _1}{S_t}{\sigma _2}{S_t}{\rm{d}}B_t^H{\rm{d}}{B_t} + {\sigma _1}{S_t}{\sigma _3}{S_t}{\rm{d}}B_t^H{\rm{d}}{P_t}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\left. { + {\sigma _2}{S_t}{\sigma _3}{S_t}{\rm{d}}B_t^{\rm{d}}{P_t}} \right]\\ \;\;\;\;\;\;\;\;\;\; = 2H\sigma _1^2S_t^2{t^{2H -1}}{\rm{d}}t + \sigma _2^2S_t^2{\rm{d}}t + \lambda \sigma _3^2S_t^2{\rm{d}}t + o\left( {{\rm{d}}t{\rm{d}}{S_t}{\rm{d}}{P_t}} \right). \end{array} $

(6)

把式(6) 代入式(5), 可得

$ \begin{array}{l} {\rm{d}}{V_t} = \frac{{\partial V}}{{\partial t}}{\rm{d}}t + \frac{{\partial V}}{{\partial S}}\left[{\mu {S_t}{\rm{d}}t + {\sigma _1}{S_t}{\rm{d}}B_t^H + {\sigma _2}{S_t}{\rm{d}}{B_t} + {\sigma _3}{S_t}{\rm{d}}{P_t}} \right]\\ \;\;\;\;\;\;\;\;\; + \frac{1}{2}\frac{{{\partial ^2}V}}{{\partial {S^2}}}\left[{2H\sigma _1^2S_t^2{t^{2H-1}}{\rm{d}}t + \sigma _2^2S_t^2{\rm{d}}t + \lambda \sigma _3^2S_t^2{\rm{d}}t} \right]\\ \;\;\;\;\;\; = \left[{\frac{{\partial V}}{{\partial t}} + \mu {S_t}\frac{{\partial V}}{{\partial S}} + \left( {H\sigma _1^2{t^{2H-1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right)S_t^2\frac{{{\partial ^2}V}}{{\partial {S^2}}}} \right]{\rm{d}}t + {\sigma _1}{S_t}\frac{{\partial V}}{{\partial S}}{\rm{d}}B_t^H\\ \;\;\;\;\;\;\;\;\; + {\sigma _2}{S_t}\frac{{\partial V}}{{\partial S}}{\rm{d}}{B_t} + {\sigma _3}{S_t}\frac{{\partial V}}{{\partial S}}{\rm{d}}{P_t} + \lambda {\rm{E}}\left[{V\left( {S\left( {1 + {j_t}} \right), t} \right)-V\left( {S, t} \right)} \right]{\rm{d}}t. \end{array} $

证毕.

2.2 分数Wick-Itô-Skorohod积分与混合跳-扩散分数随机微分方程

定义2^[13] 定义函数$Y$: ${\rm {\bf R}}\to (\delta )^\ast $, $Y_t \diamondsuit B_t^H $在$(\delta )^\ast $下$\mathrm {d}t$可积, 则$Y_t $关于$B_t^H $的分数Wick-Itô-Skorohod积分定义为

$ \int_{\bf{R}} {{Y_t}{\rm{d}}B_t^H} : = \int_{\bf{R}} {{Y_t}\diamondsuit B_t^H{\rm{d}}t}, $

其中$\diamondsuit $表示Wick乘积.

引理8^[13](分数Girsanov公式Ⅰ) 假设$\psi \in L^p(P^H)$, $p>1$, 设$\gamma \in L_\phi ^2 ({\rm {\bf R}})\cap C({\rm {\bf R}})\subset \delta'({\rm {\bf R}})$, 函数$\phi $在${\bf R}$上可积, 积分表示为$\widetilde {\gamma }=\int_{\rm {\bf R}} {\phi (t, s)} \mathrm {d}s$, 定义$L^\rho(P^H)$($\rho < p$)上的映射$\omega \mapsto \psi (\omega +\widetilde {\gamma })$, 使得$\psi(\omega)\cdot\exp\lozenge(\langle\omega, \gamma\rangle)$在$\delta'({\bf R})$中可积, 则

$ \int_{\delta '\left( {\bf{R}} \right)} {\psi \left( {\omega + \tilde \gamma } \right){\rm{d}}{P^H}\left( \omega \right)} = \int_{\delta '\left( {\bf{R}} \right)} {\psi \left( \omega \right) \cdot {\rm{exp}}\diamondsuit \left( {\left\langle {\omega, \gamma } \right\rangle } \right){\rm{d}}{P^H}\left( \omega \right)}, $

其中$\langle\omega, \gamma\rangle$表示定义在$\delta(\bf R)$上的内积

$ \left\langle {\omega, \gamma } \right\rangle : = \int_{\bf{R}} {\int_{\bf{R}} {\omega \left( s \right)\gamma \left( t \right)\varphi \left( {s, t} \right){\rm{d}}s{\rm{d}}t} }, \;\;\;\;\omega, \gamma \in \delta \left( {\bf{R}} \right). $

引理8^*[13](分数Girsanov公式Ⅱ) 设$\gamma$是一个连续函数, 且supp$\gamma\subset[0, T]$, $K$是一个函数, 且supp$K\subset[0, T]$, 则

$ {\left\langle {K, f} \right\rangle _H} = {\left\langle {\gamma, f} \right\rangle _{{L^2}\left( {\bf{R}} \right)}}, {\rm{supp}}\;f \subset \delta \left( {\bf{R}} \right). $

令$\gamma (t) = \int {_{\bf{R}}} K(s)\varphi (s,t){\rm{d}}s,0 \le t \le T,F_t^H$为分数布朗运动$\{B_s^H:0\leq s\leq T\}$生成的$\sigma$-代数流, 定义概率测度$P^{H, \gamma}$, 满足$\mathrm d P^{H, \gamma}/\mathrm d P^H=\exp\lozenge(-\langle\omega, K\rangle)$, 于是在概率测度$P^{H, \gamma}$下${\hat B^H}(t) = B_t^H + \int_0^t {{\gamma _s}{\rm{d}}s} ,0 \le t \le T $, 是一个分数布朗运动.

引理9^[13](不同白噪声空间上的Wick乘积) 设$P=P^H$, $Q=P^{H, \gamma }$, $\widehat {B}^H(t)=B_t^H +\int_0^t {\gamma_s } \mathrm {d}s$, $P$和$Q$在相应的概率测度$P^H$和$P^{H, \gamma }$下的Wick乘积表示为$\diamondsuit_P $和$\diamondsuit _Q $, 则

$ F{\diamondsuit _P}G = F{\diamondsuit _Q}G, $

其中$F, G\in (\delta )_H^\ast $.

假设无风险利率的债券价格$B_t $满足

$ \left\{ \begin{array}{l} {\rm{d}}{S_t} = {r_t}{S_t}{\rm{d}}t, \\ {S_0} = 1. \end{array} \right. $

(7)

根据定义2, 引理8和引理9, 可以考虑如下混合跳-扩散分数随机微分方程

$ \left\{ \begin{array}{l} {\rm{d}}{S_t} = {S_t}\diamondsuit \left[{\left( {r-q-\lambda {\sigma _3}} \right){\rm{d}}t + {\sigma _1}{\rm{d}}B_t^H + {\sigma _2}{\rm{d}}{B_t} + {\sigma _3}{\rm{d}}{P_t}} \right], \\ {S_t} = S, \end{array} \right. $

其中$r$是期望收益率, $q$是股息分红率, $\sigma _1, \sigma _2, \sigma _3 $都是常数, 且$B_t $, $B_t^H $, $P_t $是相互独立的.作为原生资产的股票是连续支付股息红利的.

设$J$是路径依赖变量, 对于到期日为$T$的回望看跌期权, 其股票价格$S_t $满足$S_t \leqslant J_t =\mathop {\max }\limits_{0\leqslant t\leqslant T} S_t $, 对于回望看涨期权, 其股票价格$S_t $满足$S_t \geqslant J_t =\mathop {\min }\limits_{0\leqslant t\leqslant T} S_t $.因此回望期权价格$V(S_t, J_t, t)$是关于$S_t, J_t, t$的函数.于是我们建立无风险投资组合策略$\prod _t =V_t -\nabla _t \diamondsuit S_t $, 选取适当的$\nabla $使得投资组合$\prod $在$(t, t+\mathrm dt)$上是无风险的, 即$\mathrm d\prod =r(V-\nabla \diamondsuit S)\mathrm dt$, 根据Itô公式可得

$ \begin{array}{l} {\rm{d}}\prod { = {\rm{d}}V- \nabla \diamondsuit {\rm{d}}S- \nabla q\diamondsuit S{\rm{d}}t} \\ \;\;\;\;\;\;\; = \frac{{\partial V}}{{\partial t}}{\rm{d}}t + \frac{{\partial V}}{{\partial J}} + \frac{1}{2}\lambda {\sigma _3}{S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}}{\rm{d}}t + \frac{{\partial V}}{{\partial S}}{\rm{d}}S + H\sigma _1^2{S^2}{t^{2H- 1}}\frac{{{\partial ^2}V}}{{\partial {S^2}}}{\rm{d}}t + \frac{1}{2}{\sigma _2}{S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}}{\rm{d}}t\\ \;\;\;\;\;\;\;\;\; + {\sigma _3}\frac{{\partial V}}{{\partial S}}{\rm{d}}{P_t} - \nabla \diamondsuit {\rm{d}}S - \nabla q\diamondsuit S{\rm{d}}t\\ \;\;\;\;\;\;\; = \left[{\frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H-1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right){S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}}-\nabla q\diamondsuit S} \right]{\rm{d}}t + \frac{{\partial V}}{{\partial J}}{\rm{d}}J\\ \;\;\;\;\;\;\;\;\; + {\sigma _3}\frac{{\partial V}}{{\partial S}}{\rm{d}}{P_t} + \left( {\frac{{\partial V}}{{\partial S}} -\nabla } \right)\diamondsuit {\rm{d}}S. \end{array} $

(8)

由于$J_t $关于$t$不可微, 记$J_n (t)=\left[{\frac{1}{t}\int_0^t {(S_\tau )^n\mathrm {d}\tau } } \right]^{\frac{1}{n}}$, 则$J_n (t)$是关于$t$的可微函数, 即$nJ_n^{n-1} (t)\frac{\mathrm dJ_n }{dt}=\frac{S_t^n -J_n^n (t)}{t}$.注意到$S_t $是关于$t$的连续函数, 则当$n\to \infty $时有$\mathop {\lim }\limits_{n\to \infty } J_n (t)=J_t $.于是式(8) 就变成

$ \begin{array}{l} d\prod = \left[{\frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H-1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right){S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}}-\nabla q\diamondsuit S} \right]{\rm{d}}t + \frac{{\partial V}}{{\partial J}}{\rm{d}}J\\ \;\;\;\;\;\;\;\;\; + {\sigma _3}\frac{{\partial V}}{{\partial S}}{\rm{d}}{P_t} + \left( {\frac{{\partial V}}{{\partial S}} - \nabla } \right)\diamondsuit {\rm{d}}S\\ \;\;\;\;\;\;\; = \left[{\frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H-1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right){S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}} + \frac{{\partial V}}{{\partial {J_n}}} \cdot \frac{{{\rm{d}}{J_n}}}{{{\rm{d}}t}}-\nabla q\diamondsuit S} \right]{\rm{d}}t\\ \;\;\;\;\;\;\;\;\; + {\sigma _3}\frac{{\partial V}}{{\partial S}}{\rm{d}}{P_t} + \left( {\frac{{\partial V}}{{\partial S}} -\nabla } \right)\diamondsuit {\rm{d}}S. \end{array} $

令$\nabla =\frac{\partial V}{\partial S}$, 则

$ \begin{array}{l} \frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H- 1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right){S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}} + \frac{{{{\left( {\frac{S}{{{J_n}}}} \right)}^{n- 1}}S- {J_n}}}{{nt}} \cdot \frac{{\partial V}}{{\partial {J_n}}} + \left( {r + q - \lambda {\sigma _3}} \right)S\frac{{\partial V}}{{\partial S}}\\ \;\;\;\;\;\;\;\; + \lambda {\rm{E}}\left[{V\left( {S\left( {1 + {j_t}} \right), t} \right)-V\left( {S, t} \right)} \right] -\left( {r + \lambda } \right)V = 0, \end{array} $

(9)

其中$S\leqslant J_n $, ${\rm E}$是$V$的期望算子.

固定$(J, t)$, 当$n\to \infty $时式(9) 中的$\frac{(\frac{S}{J_n })^{n-1}S-J_n }{nt}\to 0$, 可得回望看跌期权的一般抛物型随机偏微分方程为

$ \begin{array}{l} \frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H- 1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right){S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}} + \left( {r- q- \lambda {\sigma _3}} \right)S\frac{{\partial V}}{{\partial S}}\\ \;\;\;\;\;\;\;\; + \lambda {\rm{E}}\left[{V\left( {S\left( {1 + {j_t}} \right), t} \right)-V\left( {S, t} \right)} \right] -\left( {r + \lambda } \right)V = 0, \end{array} $

(10)

$ V\left( {t, S, J} \right) = J-S, $

(11)

$ \frac{{\partial V}}{{\partial J}}\left| {_{S = J}} \right. = 0. $

(12)

3 主要结果 3.1 欧式固定履约回望期权和欧式浮动履约回望期权

回望期权是指该期权持有者在期权到期日可以观察期权有效期内标的资产价格的变化过程, 通过选择资产价格的最高价格或最低价格进行交易, 是一种典型复杂的新型奇异期权.设$T$为回望周期$[T_0, T]$的终止时间, 风险资产的价格$S_t $在$t\;(0\leqslant t\leqslant T)$时刻取得的最大值和最小值分别表示为^[14]

$ M_{{T_0}}^T = \mathop {\max }\limits_{{T_0} \le \xi \le T} {S_\xi } $

和

$ m_{{T_0}}^T = \mathop {\min }\limits_{{T_0} \le \xi \le T} {S_\xi }. $

回望期权有固定履约回望期权和浮动履约回望期权两类.浮动履约回望看涨期权是指该期权持有者在回望期内有权利以最低的交易价格买入资产, 浮动履约回望看跌期权是指该期权持有者在回望期内有权利以最高的交易价格买出资产, 因此$S_t \geqslant m_{T_0 }^T $或者$S_t \leqslant M_{T_0 }^T $, 到期实际支付额看涨或看跌分别为$S_t -m_{T_0 }^T, \; M_{T_0 }^T -S_t $.而对于固定履约回望期权是提前约定敲定价格$K$, 固定履约看涨或看跌实际支付额分别为$M_{T_0 }^T -K, \;K-m_{T_0 }^T $.

引理10^[14] 假设一支欧式固定履约回望看涨期权的到期实际支付额为$M_{T_0 }^T -K$, 设$S_t =S, \;M_{T_0 }^T =M$, $Q$为风险中性测度, 则在任意$t\;(T_0 \leqslant t\leqslant T)$时刻回望看涨期权价格可表示为

$ \begin{array}{l} {c_{{\rm{fixed}}}}\left( {K, M, t} \right) = \\ \left\{ \begin{array}{l} \exp \left\{ {- r\left( {T- t} \right)} \right\}{E_Q}\left[{\max \left( {M_t^T-K, 0} \right)} \right], M \le K\\ \exp \left\{ { - r\left( {T - t} \right)} \right\}\left( {M - K} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}{E_Q}\left[{\max \left( {M_t^T-K, 0} \right)} \right], M > K \end{array} \right.\\ = \exp \left\{ { - r\left( {T - t} \right)} \right\}\max \left( {M - K, 0} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}{E_Q}\left[{\max \left( {M_t^T-K, 0} \right)} \right], \end{array} $

同理, 欧式固定履约回望看跌期权的到期实际支付额为$K-m_{T_0 }^T $时, 设$m_{T_0 }^T =m$, 则在任意$t\, (T_0 \leqslant t\leqslant T)$时刻回望看跌期权价格可表示为

$ {p_{{\rm{fixed}}}}\left( {K, m, t} \right) = \exp \left\{ {- r\left( {T- t} \right)} \right\}\max \left( {M- m, 0} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}{E_Q}\left[{\max \left( {M-m, 0} \right)} \right]. $

引理11^[14] 假设一支欧式浮动履约回望看涨期权的到期实际支付额为$S_T -m_{T_0 }^T $, 设$S_t =S, \;m_{T_0 }^T =m$, , $Q$为风险中性测度, 则在任意$t\, (T_0 \leqslant t\leqslant T)$时刻回望看涨期权价格可表示为

$ \begin{array}{l} {c_{{\rm{floating}}}}\left( {{S_T}, m, t} \right) = \exp \left\{ {- r\left( {T- t} \right)} \right\}{E_Q}\left[{{S_T}-\min \left( {m, m_t^T} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \exp \left\{ { - r\left( {T - t} \right)} \right\}{E_Q}\left[{\left( {{S_T}-m} \right) + \max \left( {m-m_t^T, 0} \right)} \right], \end{array} $

当欧式浮动履约回望看跌期权的到期实际支付额为$M_{T_0 }^T -S_T $时, 设$M_{T_0 }^T =M$, 则在任意$t\, (T_0 \leqslant t\leqslant T)$时刻回望看跌期权价格可表示为

$ \begin{array}{l} {p_{{\rm{floating}}}}\left( {{S_T}, M, t} \right) = \exp \left\{ {- r\left( {T- t} \right)} \right\}{E_Q}\left[{\max \left( {M, M_t^T} \right)-{S_T}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \exp \left\{ { - r\left( {T - t} \right)} \right\}{E_Q}\left[{\max \left( {M_t^T-M, 0} \right)-\left( {{S_T}-M} \right)} \right]. \end{array} $

3.2 定价公式

设$V(t, S, J)$表示回望期权在任意$t\;(T_0 \leqslant t\leqslant T)$时刻的价格, 根据2.2小节的讨论, 欧式回望期权满足如下抛物型随机偏微分方程

$ \begin{array}{l} 0 = \frac{{\partial V}}{{\partial t}} + \left( {H\sigma _1^2{t^{2H- 1}} + \frac{1}{2}\sigma _2^2 + \frac{1}{2}\lambda \sigma _3^2} \right){S^2}\frac{{{\partial ^2}V}}{{\partial {S^2}}} + \left( {r- q- \lambda {\sigma _3}} \right)S\frac{{\partial V}}{{\partial S}} - \left( {r + \lambda } \right)V\\ \;\;\;\;\; + \left\{ \begin{array}{l} \lambda {\rm{E}}\left[{V\left( {S\left( {1 + {j_t}} \right), \min \left\{ {J, S\left( {1 + {j_t}} \right)} \right\}, t} \right)} \right], 回望看涨, \\ \lambda {\rm{E}}\left[{V\left( {S\left( {1 + {j_t}} \right), \max \left\{ {J, S\left( {1 + {j_t}} \right)} \right\}, t} \right)} \right], 回望看跌, \end{array} \right. \end{array} $

(13)

$ V\left( {t, S, J} \right) = J-S, $

(14)

其中$T_0 \leqslant t\leqslant T$, 并且$V(t, S, J)$在如下区域连续可微.

$ \sum { = \left\{ \begin{array}{l} \left\{ {\left( {S, J} \right):0 < J \le S < \infty } \right\}, 回望看涨, \\ \left\{ {\left( {S, J} \right):0 < S \le J < \infty } \right\}, 回望看跌. \end{array} \right.} $

(15)

终端条件为

$ \frac{{\partial V}}{{\partial J}}\left| {_{S = J}} \right. = 0. $

(16)

类似于引理10和引理11的推导, 在(13)-(16) 的条件下, 下面以定理的形式给出欧式固定履约回望期权和浮动履约回望期权的定价公式.

定理1 设到期日为$T$, 对于欧式固定履约回望期权, 在模型(13) 的环境下欧式固定履约回望看跌和看涨期权的价格为

$ \begin{array}{l} {P_{{\rm{fixed}}}}\left( {K, m, t} \right) = \exp \left\{ {- r\left( {T- t} \right)} \right\}\left( {K- m} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}KN\left( { - {d_2}} \right) - \exp \left\{ {\left( { - q - \lambda {\sigma _3}} \right)} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { \times \left( {T - t} \right)} \right\}mN\left( { - {d_1}} \right) + \frac{r}{{2\left( { - q - {\lambda _3}} \right)}}\left[{\exp \left\{ {\left( {-q-\lambda {\sigma _3}} \right)\left( {T-t} \right)} \right\}mN\left( {{d_1}} \right)} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { - \exp \left\{ { - r\left( {T - t} \right)} \right\}\exp \left\{ {\frac{{2K\left( { - q - \lambda {\sigma _3}} \right)}}{r}} \right\}N\left( {d_1^r} \right)} \right], \end{array} $

(17)

$ \begin{array}{l} {C_{{\rm{fixed}}}}\left( {K, m, t} \right) = \exp \left\{ {- r\left( {T- t} \right)} \right\}\left( {K- m} \right) + \exp \left\{ {\left( { - q - \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right\}MN\left( {{d_1}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \exp \left\{ { - r\left( {T - t} \right)} \right\}KN\left( { - {d_2}} \right) + \frac{r}{{2\left( { - q - \lambda {\sigma _3}} \right)}}\left[{\exp \left\{ {-r\left( {T-t} \right)} \right\}} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { \times \exp \left\{ {\frac{{2K\left( {-q - \lambda {\sigma _3}} \right)}}{r}} \right\}N\left( {d_1^r} \right) - \exp \left\{ {\left( { - q - \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right\}N\left( {{d_1}} \right)} \right\}, \end{array} $

(18)

其中

$ \begin{array}{l} {d_1} = \\ \frac{{\ln \frac{{S\sum\limits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ -1}}{\varepsilon _n}} }}{K} + \left( {r -q -\lambda {\sigma _3}} \right)\left( {T - t} \right) + \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \frac{1}{2}\left( {\sigma _2^2 + \lambda \sigma _3^2} \right)\left( {T - t} \right)}}{{\sqrt {2\sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \sigma _2^2\left( {T - t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} }}, \end{array} $

$ \begin{array}{l} {d_2} = \\ \frac{{\ln \frac{{S\sum\limits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ -1}}{\varepsilon _n}} }}{K} -\left( {r -q - \lambda {\sigma _3}} \right)\left( {T - t} \right) + \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \frac{1}{2}\left( {\sigma _2^2 + \lambda \sigma _3^2} \right)\left( {T - t} \right)}}{{\sqrt {2\sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \sigma _2^2\left( {T - t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} }}, \end{array} $

$ d_1^r = {d_1}-\frac{{2\left( {-q-\lambda {\sigma _3}} \right)\left( {T - t} \right)}}{{\sqrt r }}, $

$\varepsilon _n $表示$\prod\limits_{i=1}^n {\ln (1+j_{t_i } )} $的期望算子, $K$为固定履约敲定价格, $N(\cdot )$为累积正态分布函数.

证 明 在风险中性金融市场, 根据引理5和引理10, 到期日为$T$的欧式固定履约回望看跌期权可以表达为

$ \begin{array}{l} {P_{{\rm{fixed}}}}\left( {K, m, t} \right) = \exp \left\{ {- r\left( {T- t} \right)} \right\}\max \left( {K- m, 0} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}{E_Q}\left[{\max \left( {K-m_t^T, 0} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \exp \left\{ { - r\left( {T - t} \right)} \right\}\max \left( {K - m, 0} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times {{\tilde E}_{{P_H}}}\left[{\min \left( {K-m_t^T, 0} \right){1_{K > m_t^T}}\left| {F_t^H} \right.} \right], \end{array} $

(19)

其中$1_{K>m_t^T } $为示性函数.

设

$ \left( {r-q-\lambda {\sigma _3}} \right)t = {\sigma _1}\left( {\hat B_T^H-\hat B_t^H} \right) + {\sigma _2}\left( {{{\hat B}_T} - {{\hat B}_t}} \right) + {\sigma _3}\left( {{{\hat P}_T} - {{\hat P}_t}} \right), $

则

$ m_t^T = {S_t} \cdot \exp \left\{ {\left[{r-q-\lambda {\sigma _3}\left( {T-t} \right)} \right] \times \left[{-\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right)-\frac{1}{2}\sigma _2^2\left( {T - t} \right) - \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right]} \right\}, $

注意到Poisson跳过程的跳跃强度为$\lambda $, 利用Poisson分布函数性质, 得

$ \begin{array}{l} {S_T} = {S_t}\prod\limits_{i = 1}^{{N_{T- t}}} {\exp \left\{ {\left[{r-q-\lambda {\sigma _3}\left( {T-t} \right)} \right] \cdot \left[{-\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right)-\frac{1}{2}\sigma _2^2\left( {T - t} \right) - \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right]} \right\}} \\ \;\;\;\; = \sum\limits_{n = 0}^\infty {{\rm{P}}\left( {{N_t} = n} \right)} \cdot \left\{ {{S_t}\prod\limits_{i = 1}^n {\exp \left[{\left( {{\sigma _1}\left( {\hat B_T^H-\hat B_t^H} \right) + {\sigma _2}\left( {{{\hat B}_T}-{{\hat B}_t}} \right) + {\sigma _3}\left( {{{\hat P}_T}-{{\hat P}_t}} \right)} \right)} \right.} } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\left. { \times \left( { - \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) - \frac{1}{2}\sigma _2^2\left( {T - t} \right) - \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right)} \right]} \right\}\\ \;\;\;\; = \sum\limits_{n = 0}^\infty {\frac{{{{\rm{e}}^{ - \lambda \left( {T - t} \right)}}{\lambda ^n}{{\left( {T - t} \right)}^n}}}{{n!}}} \left\{ {{S_t}\prod\limits_{i = 1}^n {\exp \left[{\left( {{\sigma _1}\left( {\hat B_T^H-\hat B_t^H} \right) + {\sigma _2}\left( {{{\hat B}_T}-{{\hat B}_t}} \right) + {\sigma _3}\left( {{{\hat P}_T}-{{\hat P}_t}} \right)} \right)} \right.} } \right.\\ \;\;\;\;\;\;\;\;\;\;\left. {\left. { \times \left( { - \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) - \frac{1}{2}\sigma _2^2\left( {T - t} \right) - \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right)} \right]} \right\}\\ \;\;\;\; = {S_t}\sum\limits_{n = 0}^\infty {\frac{{{{\rm{e}}^{ - \lambda \left( {T - t} \right)}}{\lambda ^n}{{\left( {T - t} \right)}^n}}}{{n!}}{\varepsilon _n}} \times \exp \left[{\left( {{\sigma _1}\left( {\hat B_T^H-\hat B_t^H} \right) + {\sigma _2}\left( {{{\hat B}_T}-{{\hat B}_t}} \right) + {\sigma _3}\left( {{{\hat P}_T}-{{\hat P}_t}} \right)} \right)} \right.\\ \;\;\;\;\;\;\;\;\;\;\left. { \times \left( { - \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) - \frac{1}{2}\sigma _2^2\left( {T - t} \right) - \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right)} \right], \end{array} $

于是

$ \begin{array}{l} m_t^T = {S_t}\sum\limits_{n = 0}^\infty {\frac{{{{\rm{e}}^{- \lambda \left( {T- t} \right)}}{\lambda ^n}{{\left( {T- t} \right)}^n}}}{{n!}}{\varepsilon _n}} \times \exp \left[{\left( {{\sigma _1}\left( {\hat B_T^H-\hat B_t^H} \right) + {\sigma _2}\left( {{{\hat B}_T}-{{\hat B}_t}} \right) + {\sigma _3}\left( {{{\hat P}_T}-{{\hat P}_t}} \right)} \right)} \right.\\ \;\;\;\;\;\;\;\;\;\;\left. { \times \left( { - \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) - \frac{1}{2}\sigma _2^2\left( {T - t} \right) - \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right)} \right], \end{array} $

对其两端取对数可得

$ \begin{array}{l} \ln m_t^T = \ln \left( {{S_t}\sum\limits_{n = 0}^\infty {\frac{{{{\rm{e}}^{- \lambda \left( {T- t} \right)}}{\lambda ^n}{{\left( {T- t} \right)}^n}}}{{n!}}{\varepsilon _n}} } \right) + \left[{{\sigma _1}\left( {\hat B_T^H-\hat B_t^H} \right) + {\sigma _2}\left( {{{\hat B}_T}-{{\hat B}_t}} \right) + {\sigma _3}\left( {{{\hat P}_T}-{{\hat P}_t}} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\; + \left[{-\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right)-\frac{1}{2}\sigma _2^2\left( {T - t} \right) - \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right]. \end{array} $

利用分数Girsanov定理可知$\sigma _1 (\widehat {B}_T^H -\widehat {B}_t^H )+\sigma _2 (\widehat {B}_T -\widehat {B}_t )+\sigma _3 (\widehat {P}_T -\widehat {P}_t )$在概率测度$\widehat {P}_H $下生成新的混合跳-扩散分数布朗运动, 记

$ \begin{array}{l} d_1^ * = \ln \frac{{S\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ - 1}}{\varepsilon _n}} }}{K} + \left( {r - q + \lambda {\sigma _3}} \right)\left( {T - t} \right)\\ \;\;\;\;\;\;\; + \left[{-\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right)-\frac{1}{2}\sigma _2^2\left( {T - t} \right) - \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right]. \end{array} $

且

$ \begin{array}{l} {d_1} = \frac{{{\sigma _1}\hat B_t^H + {\sigma _2}{{\hat B}_t} + {\sigma _3}{{\hat P}_t}- d_1^ * }}{{\sqrt {2\sigma _1^2\left( {{T^{2H}}- {t^{2H}}} \right) + \sigma _2^2\left( {T- t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} }}\\ \;\;\;\; = \left\{ {\ln \frac{{S\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T - t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ - 1}}{\varepsilon _n}} }}{K} + \left( {r - q + \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right.\\ \;\;\;\;\left. { + \left[{\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right) + \frac{1}{2}\sigma _2^2\left( {T-t} \right) + \frac{1}{2}\lambda \sigma _3^2\left( {T-t} \right)} \right]} \right\}\\ \;\;\;\; \times {\left\{ {\sqrt {2\sigma _1^2\left( {{T^{2H}} -{t^{2H}}} \right) + \sigma _2^2\left( {T -t} \right) + \lambda \sigma _3^2\left( {T -t} \right)} } \right\}^{ - 1}}, \end{array} $

根据引理5, 可得

$ \begin{array}{l} {{\tilde E}_{{{\hat P}_H}}}\left[{{1_{K > m_t^T}}\left| {F_t^H} \right.} \right] = {{\tilde E}_{{{\hat P}_H}}}\left[{{1_{d_1^ * < {\sigma _1}\hat B_t^H + {\sigma _2}{{\hat B}_T} + {\sigma _3}{{\hat P}_T} < + \infty }}\left| {F_t^H} \right.} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \int_{d_1^ * }^{ + \infty } {\frac{1}{{\sqrt {2\pi \left[{2\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right) + \sigma _2^2\left( {T-t} \right) + \lambda \sigma _3^2\left( {T-t} \right)} \right]} }}} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left[{-\frac{{{{\left( {x-{\sigma _1}\hat B_t^H-{\sigma _2}{{\hat B}_t} - {\sigma _3}\hat P} \right)}^2}}}{{2\sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \sigma _2^2\left( {T - t} \right) + \lambda \sigma _3^2\left( {T - t} \right)}}} \right]{\rm{d}}x\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \int_{ -\infty }^{{d_1}} {\frac{1}{{\sqrt {2\pi } }}\exp \left( { -\frac{{{z^2}}}{2}} \right){\rm{d}}z} = N\left( {{d_1}} \right). \end{array} $

(20)

考察$\widetilde {E}_{\widehat {P}_H } [m_t^T 1_{K>m_t^T } \vert F_t^H]$, 记

$ {\sigma _1}{\left( {B_t^H} \right)^ * } + {\sigma _2}B_t^ * + {\sigma _3}P_t^ * = {\sigma _1}\left( {\hat B_t^H-{\sigma _1}t} \right) + {\sigma _2}\left( {{{\hat B}_t}-{\sigma _2}t} \right) + {\sigma _3}\left( {{{\hat P}_t}-{\sigma _3}t} \right), $

再次利用分数Girsanov定理可知$\sigma _1 (B_t^H )^\ast +\sigma _2 B_t^\ast +\sigma _3 P_t^\ast $在概率测度$P_H^\ast $下生成新的混合跳-扩散分数布朗运动, 设

$ {X_t} = \exp \left( {{\sigma _1}\hat B_t^H + {\sigma _2}{{\hat B}_t} + {\sigma _3}{{\hat P}_t}-\sigma _1^2{t^{2H}}-\frac{1}{2}\sigma _2^2t-\frac{1}{2}\lambda \sigma _3^2t} \right), $

则$m_t^T =S_t \exp \{(r-q-\lambda \sigma _3 )(T-t)\}X_T $, 根据引理4和引理5可得

$ \begin{array}{l} \;\;\;{{\tilde E}_{{{\hat P}_H}}}\left[{m_t^T{1_{K > m_t^T}}\left| {F_t^H} \right.} \right]\\ = {S_t}\exp \left\{ {\left( {r - q - \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right\}{{\tilde E}_{{{\hat P}_H}}}\\ \left[{{X_T}{1_{d_1^ * < {\sigma _1}\hat B_t^H + {\sigma _2}{{\hat B}_T} + {\sigma _3}{{\hat P}_T} < + \infty }}\left( {{\sigma _1}\hat B_T^H + {\sigma _2}{{\hat B}_T} + {\sigma _3}{{\hat P}_T}} \right)\left| {F_t^H} \right.} \right]\\ = {S_t}\exp \left\{ {\left( {r - q - \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right\}{{\tilde E}_{P_H^ * }}\\ \left[{{1_{d_2^ * < \left[{{\sigma _1}{{\left( {\hat B_T^H} \right)}^ * } + {\sigma _2}\hat B_T^ * + {\sigma _3}\hat P_T^ * } \right] < + \infty }}\left( {{\sigma _1}{{\left( {\hat B_T^H} \right)}^ * } + {\sigma _2}\hat B_T^ * + {\sigma _3}\hat P_T^ * } \right)\left| {F_t^H} \right.} \right]\\ = {S_t}\exp \left\{ {\left( {r -q -\lambda {\sigma _3}} \right)\left( {T -t} \right)} \right\} \times \int_{ - \infty }^{{d_2}} {\frac{1}{{\sqrt {2\pi } }}\exp \left( { - \frac{{{z^2}}}{2}} \right){\rm{d}}z} \\ = {S_t}\exp \left\{ {\left( {r - q - \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right\} \times N\left( {{d_2}} \right), \end{array} $

(21)

其中

$ \begin{array}{l} {d_2} = \frac{{{\sigma _1}{{\left( {\hat B_t^H} \right)}^ * } + {\sigma _2}\hat B_t^ * + {\sigma _3}\hat P_t^ *- d_2^ * }}{{\sqrt {2\sigma _1^2\left( {{T^{2H}}- {t^{2H}}} \right) + \sigma _2^2\left( {T- t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} }}\\ \;\;\;\; = \left\{ {\ln \frac{{S\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T - t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ - 1}}{\varepsilon _n}} }}{K} - \left( {r - q + \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right.\\ \;\;\;\;\left. { + \left[{\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right) + \frac{1}{2}\sigma _2^2\left( {T-t} \right) + \frac{1}{2}\lambda \sigma _3^2\left( {T-t} \right)} \right]} \right\}\\ \;\;\;\; \times {\left\{ {\sqrt {2\sigma _1^2\left( {{T^{2H}} -{t^{2H}}} \right) + \sigma _2^2\left( {T -t} \right) + \lambda \sigma _3^2\left( {T -t} \right)} } \right\}^{ - 1}}, \end{array} $

$ \begin{array}{l} d_2^ * = \ln \frac{{S\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ - 1}}{\varepsilon _n}} }}{K} - \left( {r - q + \lambda {\sigma _3}} \right)\left( {T - t} \right)\\ \;\;\;\;\;\;\; + {\sigma _1}{\left( {B_t^H} \right)^ * } + {\sigma _2}B_t^ * + {\sigma _3}P_t^ * + \left[{\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right) + \frac{1}{2}\sigma _2^2\left( {T-t} \right) + \frac{1}{2}\lambda \sigma _3^2\left( {T-t} \right)} \right]. \end{array} $

根据式(19)-(21), 以及引理10, 可得

$ \begin{array}{l} \;\;\;{P_{{\rm{fixed}}}}\left( {K, m, t} \right)\\ = \exp \left\{ {- r\left( {T- t} \right)} \right\}\max \left( {K- m, 0} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}{{\tilde E}_{{P_H}}}\left[{\min \left( {K-m_t^T, 0} \right){1_{K > m_t^T}}\left| {F_t^H} \right.} \right] \end{array} $

$ \begin{array}{l} = \exp \left\{ {- r\left( {T- t} \right)} \right\}{{\tilde E}_{{P_H}}}\left[{\max \left( {K-m, 0} \right){1_{K > m}}\left| {F_t^H} \right.} \right] + \exp \left\{ { -r\left( {T -t} \right)} \right\}\\ \times \int_0^{ + \infty } {{\rm{P}}\left( {\max \left( {K -m_t^T, 0} \right) \ge x} \right){\rm{d}}x}\\ \left( {0 \le \max \left( {K - m, 0} \right) \le K, 0 \le \max \left( {K - m_t^T, 0} \right) \le K} \right) \end{array} $

$ = \exp \left\{ {-r\left( {T-t} \right)} \right\}\left( {K-m} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}\int_0^K {{\rm{P}}\left( {K - m_t^T \ge x} \right){\rm{d}}x} $

$ = \exp \left\{ {-r\left( {T-t} \right)} \right\}\left( {K-m} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}\int_0^K {{\rm{P}}\left( {\ln \frac{{m_t^T}}{m} \le \ln \frac{z}{m}} \right){\rm{d}}z} \left( {z = K - z} \right) $

$ = \exp \left\{ {-r\left( {T-t} \right)} \right\}\left( {K-m} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}\int_0^{\ln \frac{K}{m}} {m\exp \left( y \right){\rm{P}}\left( {\ln \frac{{m_t^T}}{m} \le y} \right){\rm{d}}y\left( {y = \ln \frac{2}{m}} \right)} $

$ \begin{array}{l} = \exp \left\{ {- r\left( {T- t} \right)} \right\}\left( {K- m} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}\int_0^{\ln \frac{K}{m}} {m\exp \left( y \right)} \\ \;\;\; \times P\left\{ {N\left[{\left( {y- \ln \frac{{m\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ -1}}{\varepsilon _n}} }}{K} + \left( {r -q -\lambda {\sigma _3}} \right)\left( {T - t} \right)} \right.} \right.} \right. \end{array} $

$ \begin{array}{l} \;\;\;\left. { + \sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right) + \frac{1}{2}\sigma _2^2\left( {T-t} \right) + \frac{1}{2}\lambda \sigma _3^2\left( {T-t} \right)} \right)\\ \;\;\;\left. { \times {{\left( {\sqrt {2\sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \sigma _2^2\left( {T - t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} } \right)}^{ - 1}}} \right] \end{array} $

$ \begin{array}{l} \;\;\; + \exp \left[{\frac{{ry}}{{2\left( {-q-\lambda {\sigma _3}} \right)}}} \right] \times N\left[{\left( {\ln \frac{{m\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ -1}}{\varepsilon _n}} }}{K}} \right.} \right.\\ \left. {\;\;\; + \left( {r -q -\lambda {\sigma _3}} \right)\left( {T - t} \right) + \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \frac{1}{2}\sigma _2^2\left( {T - t} \right) + \frac{1}{2}\lambda \sigma _3^2\left( {T - t} \right)} \right) \end{array} $

$ \begin{array}{l} \left. {\left. {\;\;\; \times {{\left( {\sqrt {2\sigma _1^2\left( {{T^{2H}}-{t^{2H}}} \right) + \sigma _2^2\left( {T-t} \right) + \lambda \sigma _3^2\left( {T-t} \right)} } \right)}^{ - 1}}} \right]} \right\}{\rm{d}}y\\ = \exp \left\{ { - r\left( {T - t} \right)} \right\}\left( {K - m} \right) + \exp \left\{ { - r\left( {T - t} \right)} \right\}KN\left( { - {d_2}} \right)\\ - \exp \left\{ {\left( { - q - \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right\}mN\left( { - {d_1}} \right)\\ + \frac{r}{{2\left( { - q - \lambda {\sigma _3}} \right)}}\\ \left[{\exp \left\{ {\left( {-q-\lambda {\sigma _3}} \right)\left( {T-t} \right)} \right\}mN\left( { - {d_1}} \right) - \exp \left\{ { - r\left( {T - t} \right)} \right\}\exp \left\{ {\frac{{2K\left( { - q - \lambda {\sigma _3}} \right)}}{r}} \right\}N\left( {d_1^r} \right)} \right], \end{array} $

其中

$ \begin{array}{*{20}{l}} {{d_1} = }\\ {\frac{{\ln \frac{{S\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ -1}}{\varepsilon _n}} }}{K} + \left( {r -q -\lambda {\sigma _3}} \right)\left( {T - t} \right) + \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \frac{1}{2}\left( {\sigma _2^2 + \lambda \sigma _3^2} \right)\left( {T - t} \right)}}{{\sqrt {2\sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \sigma _2^2\left( {T - t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} }}, } \end{array} $

$ \begin{array}{*{20}{l}} {{d_2} = }\\ {\frac{{\ln \frac{{S\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ -1}}{\varepsilon _n}} }}{K} -\left( {r -q - \lambda {\sigma _3}} \right)\left( {T - t} \right) + \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \frac{1}{2}\left( {\sigma _2^2 + \lambda \sigma _3^2} \right)\left( {T - t} \right)}}{{\sqrt {2\sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \sigma _2^2\left( {T - t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} }}, } \end{array} $

$ d_1^r = {d_1}-\frac{{2\left( {-q-\lambda {\sigma _3}} \right)\left( {T - t} \right)}}{{\sqrt r }}. $

利用同样的证明过程可以得到式(18).

对照引理11以及定理1, 可以给出欧式浮动履约回望看跌和看涨期权的价格公式.

定理2 设到期日为$T$, 对于欧式浮动履约回望期权, 在模型(13) 的环境下欧式浮动履约回望看涨和看跌期权的价格为

$ \begin{array}{l} {C_{{\rm{floating}}}}\left( {{S_T}, m, t} \right) = \\ \exp \left\{ {- r\left( {T- t} \right)} \right\}{S_T}N\left( {- {d_2}} \right) - \exp \left\{ {\left( { - q - \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right\}mN\left( { - {d_1}} \right) + \frac{r}{{2\left( { - q - \lambda {\sigma _3}} \right)}}\\ \times \left[{\exp \left\{ {\left( {-q-\lambda {\sigma _3}} \right)\left( {T-t} \right)} \right\}mN\left( {{d_1}} \right) - \exp \left\{ { - r\left( {T - t} \right)} \right\}\exp \left\{ {\frac{{2{S_T}\left( { - q - \lambda {\sigma _3}} \right)}}{r}} \right\}N\left( {d_1^r} \right)} \right], \end{array} $

(22)

$ \begin{array}{l} {P_{{\rm{floating}}}}\left( {{S_T}, M, t} \right) = \\ \exp \left\{ {\left( {- q- \lambda {\sigma _3}} \right)\left( {T- t} \right)} \right\}MN\left( { - {d_1}} \right) - \exp \left\{ { - r\left( {T - t} \right)} \right\}{S_T}N\left( { - {d_2}} \right) + \frac{r}{{2\left( { - q - \lambda {\sigma _3}} \right)}}\\ \times \left[{\exp \left\{ {-r\left( {T-t} \right)} \right\}\exp \left\{ {\frac{{2{S_T}\left( {-q - \lambda {\sigma _3}} \right)}}{r}} \right\}N\left( {d_1^r} \right) - \exp \left\{ {\left( { - q - \lambda {\sigma _3}} \right)\left( {T - t} \right)} \right\}N\left( {{d_1}} \right)} \right], \end{array} $

(23)

$ \begin{array}{*{20}{l}} {{d_1} = }\\ {\frac{{\ln \frac{{S\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ -1}}{\varepsilon _n}} }}{{{S_T}}} + \left( {r -q -\lambda {\sigma _3}} \right)\left( {T - t} \right) + \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \frac{1}{2}\left( {\sigma _2^2 + \lambda \sigma _3^2} \right)\left( {T - t} \right)}}{{\sqrt {2\sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \sigma _2^2\left( {T - t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} }}, } \end{array} $

$ \begin{array}{*{20}{l}} {{d_2} = }\\ {\frac{{\ln \frac{{S\sum\nolimits_{n = 0}^\infty {{\lambda ^n}{{\left( {T- t} \right)}^n}\exp \left[{-\lambda \left( {T-t} \right)} \right]{{\left( {n!} \right)}^{ -1}}{\varepsilon _n}} }}{K} -\left( {r -q - \lambda {\sigma _3}} \right)\left( {T - t} \right) + \sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \frac{1}{2}\left( {\sigma _2^2 + \lambda \sigma _3^2} \right)\left( {T - t} \right)}}{{\sqrt {2\sigma _1^2\left( {{T^{2H}} - {t^{2H}}} \right) + \sigma _2^2\left( {T - t} \right) + \lambda \sigma _3^2\left( {T - t} \right)} }}, } \end{array} $

$ d_1^r = {d_1}-\frac{{2\left( {-q-\lambda {\sigma _3}} \right)\left( {T - t} \right)}}{{\sqrt r }}, $

其中$\varepsilon _n $表示$\prod\limits_{i=1}^n {\ln (1+j_{t_i } )} $的期望算子, $S_T $为浮动履约敲定价格, $N(\cdot )$为累积正态分布函数.

证明过程类似于定理1.

注记1 根据定理1和定理2, 欧式固定履约回望看涨期权和看跌期权之间满足如下的平价关系

$ {C_{{\rm{fixed}}}}\left( {K, M, t} \right) = {P_{{\rm{floating}}}}\left( {{S_T}, M, t} \right) + \exp \left\{ {-r\left( {T-t} \right)} \right\}\max \left( {M-K, 0} \right). $

Eberlein E和Papapantoleon A^[16]也得到了风险资产在Lévy跳过程的环境下欧式固定履约回望看涨期权和欧式浮动履约回望看跌期权之间的平价关系公式.

注记2 在本文的混合跳-扩散分数布朗运动模型中, 当$H=\frac{1}{2}$时, 模型退化成了标准跳-扩散布朗运动模型, 是本文模型的特殊情形, 不再讨论.

4 数值模拟及分析 4.1 几种定价模型的对比

对不含有跳过程的纯混合分数布朗运动(pmfBm)模型, 含有跳过程的混合分数布朗运动(jmfBm)模型^[7], 以及本文的混合跳-扩散分数布朗运动(mj-dfBm)模型分别作一般假设下的理论价格对比, 为了更突出模型之间的异同, 参数的选择没有选取已有的实证数据, 所有模拟结果由Matlab运行生成, 模型所用到的所有参数值见表 1.

把表 1的相应参数分别代入到各类模型中, 记$P_{p-m} $, $P_{j-m} $, $P_{m-j} $分别为pmfBm模型, jmfBm模型, mj-dfBm模型下欧式固定履约回望看跌期权价格, 计算结果见表 2.

比较$P_{p-m} $, $P_{j-m}^{\ast, {l}} $, $P_{j-m}^{\ast, {h}} $, $P_{m-j}^{\ast, {l}} $, $P_{m-j}^{\ast, {h}} $的各列价格值, 可以发现欧式固定履约回望看跌期权价格的变化是一致的, 都是随着持有货币比率$(m/K)$的递增而递减.在参数的选择比较小的情况下, jmfBm模型和mj-dfBm模型在低强度跳和高强度跳时的价格比较接近.最大的不同处在于, mj-dfBm模型在同等条件下价格的递减速率明显要快于其他模型.

4.2 相关参数对模型的影响

考虑混合跳-扩散分数布朗运动下欧式固定履约回望看跌期权定价模型各参数的变化, 在取不同的跳参数值时, 相应的$H$, $\lambda $, $\mu _j $, $\sigma _j $参数值的变化情况如图 1所示.这里边界参数设定为$r=0. 0250$, $q=0. 0320$, $\sigma _1 =0. 1073, K=100, $$H=0. 558, J=0. 0035, $$t=0, T=2, \lambda =7. 68, $$\mu _j =-000721, \sigma _j =0. 0019, $$\sigma _2 =0. 0215, \sigma _3 =0. 0007202. $ 图 1表明, 回望看跌期权价格随着参数$H$, $\lambda $, $\mu _j $, $\sigma _j $的单调增加而单调减少.类似的单调现象也出现在文献[12, 15]中.

继续考虑混合跳扩散分数布朗运动下欧式固定履约回望看跌期权定价模型的隐含波动率变化, 在选定参数$r=0. 0250$, $q=0. 0320$, $\sigma _1 =0. 1073$, $K=100$, $H=0. 558$, $J=0. 0035$, $t=0$, $T=0. 5$(年), $\lambda =10. 22$, $\mu _j =-000721, \sigma _j =0. 0019, \sigma _2 =0. 0215, \sigma _3 =0. 0007202$时, 绘制曲面图形如图 2所示.很显然, 随着持有货币比率$(m/K)$的减少, 在回望期权的有效回望期内, 模型对回望期权的隐含波动率表现出明显的“微笑”现象.

5 结论

本文的定价模型考虑了标准布朗运动、分数布朗运动、Poisson过程的线性组合, 利用Itô公式和分数Wick-Itô-Skorohod积分建立了一个新的市场定价模型, 给出了连续支付红利的欧式固定履约和浮动履约回望期权的定价公式, 通过具体数值和相关参数对定价公式的模拟验证, 表明该模型在刻画期权在短期交割或长期交割前表现出的异常波动跳现象是相当合理的并且是有效性的, 从回望看跌期权价格的递减表现上来看, 明显略快于其他模型, 更加接近不确定金融市场.本文的混合跳-扩散分数布朗运动模型便于计算, 可操作性强, 不但能有效地刻画金融市场的隐含波动率的变化, 而且能够合理地解释金融市场的“微笑”现象.本文的结论在合理解释金融现象的同时, 也丰富了期权定价理论, 对其他类型奇异期权的研究以及一些复杂金融衍生品定价的研究具有参考意义.