近些年来,用分数布朗运动来研究金融数学模型已经成为一个重要的发展方向.由于分数布朗运动的“尖峰厚尾性”和“长程记忆性”表现出Gussian分布的特性,所以用Gussian分布能够更好的刻画金融资产模型,相应的随机分析理论见文献[1-3].然而, Tomas^[4]]等人的研究表明,在使用分数布朗运动来刻画金融资产价格的波动时仍存在一些不足,基于Wick积分的分数布朗运动在应用于金融计算时受到很大的限制,同时定义一个合适的关于分数布朗运动的随机积分是比较困难的.于是,考虑使用混合分数布朗运动来刻画金融资产的波动是合理的(如Xiao^[5], Sun^[6], He^[7]等). Cheridito^[8]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; Mounir^[9]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;由于分数布朗运动的Itô公式所建立的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^[10].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,于是有些学者提出用混合跳扩散分数布朗运动(mjd-fBm)模型来刻画金融市场的波动行为(如Foad^{[11, 12]}, Rao^[13], Miao^[14], Yang^[15]等).

关于具有复杂收益结构的奇异型金融衍生证券定价的研究成果并不多见,本文拟通过一个基于标准布朗运动、分数布朗运动、Poisson过程的线性组合的混合跳扩散B-S模型,结合Merton假设条件以及风险资产所满足的随机微分方程的Cauchy初值问题,利用多尺度参数摄动方法(张伟江^[16], Nesterov^[17], Ma^[18], Butuzov^[19]等),求解了欧式回望期权所适合的抛物型随机偏微分方程,给出了欧式固定履约价的回望期权的近似定价公式,并利用Feymann-Kac公式分析了近似公式的误差估计.结合已有的成果(Lai^[20], Eberlein^[21], Leung^[22], Park^[23], Fuh^[24],杨朝强^{[25, 26]}等),文章最后对近似公式作了数值模拟分析.

定义2.1^[8]  设Hurst参数为$ H $的分数布朗运动$ \{B_t^H\}_{t\geq0} $是一个连续Gussian过程,且$ H\in(0, 1) $,定义含有参数$ \alpha, \beta $和$ H $的混合分数布朗运动$ M_t^H $,是由Hurst参数$ 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, \mathfrak{F}, \mathbb{P}) $,对任意的$ t\in\mathbb{R}^+ $,满足

$ \begin{array}{lll} M_t^H = \alpha B_t+\beta B_t^H, \end{array} $

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)\neq(0, 0) $.

性质2.1^[9]  混合分数布朗运动$ M_t^H $具有如下性质

(ⅰ)对任意的$ H \in(0, 1)\setminus 1/2 $, $ M_t^H $是中心高斯过程;

(ⅱ) $ M_0^H = 0 $, ($ \mathbb{P} $-a.s.);

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

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

其中$ \wedge $表示两个数中取最小;

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

$ \begin{eqnarray*} M_{ht}^H(\alpha, \beta)\dot{ = }M_t^H(\alpha h^{\frac{1}{2}}, \beta h^H), \end{eqnarray*} $

其中$ \dot{ = } $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 \mathbb{R}^+ $, $ M_t^H(\alpha, \beta) $满足

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

Cheridito^[8]已经证明,当Hurst参数$ H\in (0, \frac{3}{4}] $时, $ M_t^H(\alpha, \beta) $不是半鞅,而当$ H\in (\frac{3}{4}, 1) $且$ \beta\neq 0 $时, $ M_t^H(\alpha, \beta) $是一个半鞅且等价于布朗运动. Bender^[10]等人证明了混合分数布朗运动市场在正则策略集中是不存在套利的,并且欧式期权均存在这样一个正则策略集将其进行套期保值,因此,市场在此策略集下是完全的.所以本文恒假设Hurst参数$ H\in (\frac{3}{4}, 1) $,并且假设市场是无摩擦的,所交易的期权只能在到期日才能被执行.

在完备概率空间$ \{\Omega, \mathfrak{F}, \mathfrak{F}_t, P\} $中考虑一个连续时间的金融市场,其中$ \{\mathfrak{F}_t\}_{0\leq t\leq 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 $是定义在空间$ (\mathfrak{F}_t, 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 $为复合Poisson过程表示为$ j_t = \sum\limits_{i = 0}^{N_t}P_i $,记在Merton假设下,有$ \ln(1+j_t)\sim\mathbb{N}[\ln(1+\mu_{j_t})-\frac{1}{2}\sigma_{j_t}^2, \sigma_{j_t}^2] $, $ \mu_{j_t} = E(P_i) = \exp\{\ln(1+j_t)-1\} $, $ \sigma_{j_t}^2 = Var\ln(1+j_t) $,显然$ \mu_{j_t} $是关于$ 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 $的有界函数,记$ \ln(1+\mu_{j_t}) = \sigma_3 $是一个常数.

引理2.1^[15]   设$ V_t = V(S_t, t) $, $ V $是二元可微函数.若动态价格过程$ S_t $适合方程

(2.1) $ \begin{equation} \mathrm{d}S_t = \mu S_t\mathrm{d}t+\sigma_1 S_t\mathrm{d}B_t^H +\sigma_2 S_t \mathrm{d}B_t+\sigma_3 S_t \mathrm{d}P_t, \end{equation} $

其中$ \mu $是期望收益率,则

$ \begin{eqnarray*} &&\mathrm{d}V_t = \left[\frac{\partial V}{\partial t}+\mu S_t \frac{\partial V}{\partial S}+\left(H\sigma_1^2 S_t^2 t^{2H-1}+\frac{1}{2}\sigma_2^2 S_t^2 +\frac{1}{2}\lambda\sigma_3^2 S_t^2 \right)\frac{\partial^2 V}{\partial S^2}\right]\mathrm{d}t +\sigma_1 S_t \frac{\partial V}{\partial S}\mathrm{d}B_t^H\\&&\; \; \; \; \; \; \; \; +\sigma_2 S_t \frac{\partial V}{\partial S}\mathrm{d}B_t+\sigma_3 S_t \frac{\partial V}{\partial S}\mathrm{d}P_t+\lambda\mathbb{E}[V(S(1+j_t), t)-V(S, t)]\mathrm{d}t, \end{eqnarray*} $

其中$ \sigma_3 S_t \frac{\partial V}{\partial S}\mathrm{d}P_t $表示Poisson跳过程$ \mathrm{d}t $时间内作用于$ \frac{\partial V}{\partial S} $的变化量, $ \lambda\mathbb{E}[V(S(1+j_t), t)-V(S, t)]\mathrm{d}t $为Poisson跳跃过程在$ \mathrm{d}t $时间内作用于$ V $的变化量, $ \mathbb{E} $表示二元可微函数$ V $的期望算子.

回望期权是一种典型复杂的新型奇异期权,期权持有者在期权到期日可通过选择资产价格在有效回望期内的最高价格或者最低价格而进行交易.设$ T $为回望周期$ [0, T] $的终止时间,风险资产的价格$ S_t $在$ t(0\leq t\leq T) $时刻取得的最大值和最小值分别表示为$ M_0^T = \max\limits_{0\leq\eta\leq T}S_\eta $和$ m_0^T = \min\limits_{0\leq\eta\leq T}S_\eta, $固定履约回望期权是指提前约定敲定价格$ K $,固定履约看涨或看跌实际支付额分别为$ M_0^T-K, K-m_0^T $.

设$ V(S, J, t) $表示回望期权在任意$ t(0\leq t\leq T) $时刻的价格($ J $是路径依赖变量),且满足如下抛物型随机偏微分方程^[15]

(2.2) $ \begin{eqnarray} 0& = &\frac{\partial V}{\partial t}+\bigg( H\sigma_1^2t^{2H-1} +\dfrac{1}{2}\sigma_2^2+\dfrac{1}{2}\lambda\sigma_3^2\bigg)S^2\dfrac{\partial^2 V}{\partial S^2}+(r-q-\lambda\sigma_3)S\dfrac{\partial V}{\partial S} -(r+\lambda)V \\&&+\left\{ \begin{aligned} &\lambda\mathbb{E}[V(S(1+j_t), \min\{J, S(1+j_t)\}, t)], \mbox{回望看涨}, \\ &\lambda\mathbb{E}[V(S(1+j_t), \max\{J, S(1+j_t)\}, t)], \mbox{回望看跌}, \end{aligned}\right. \end{eqnarray} $

其中$ 0\leq t\leq T $,并且$ V(S, J, t) $在如下区域连续可微

(2.3) $ \begin{equation} \Sigma = \left\{ \begin{aligned} & \{(S, J):0<J\leq S<\infty\}, \mbox{回望看涨}, \\ &\{(S, J):0<S\leq J<\infty\}, \mbox{回望看跌}, \end{aligned} \right. \end{equation} $

终端条件为

(2.4) $ \begin{equation} \frac{\partial V}{\partial J}\Big|_{S = J} = 0. \end{equation} $

注意到抛物型随机偏微分方程(2.2)的结构复杂,寻找回望期权的精确解析解是比较困难的^[25],其解的结构相当复杂.于是,考虑在混合跳扩散B-S模型下研究回望期权的近似解,不妨设原生资产价格$ S_t $满足如下随机微分方程

(2.5) $ \begin{eqnarray} \mathrm{d}S_t& = &[\mu(t, S_t)-q-\lambda\sigma_3(t, S_t;\varepsilon_3)] S_t\mathrm{d}t \\&&+\sigma_1(t, S_t;\varepsilon_1) S_t\mathrm{d}B_t^H +\sigma_2(t, S_t;\varepsilon_2) S_t \mathrm{d}B_t+\sigma_3(t, S_t;\varepsilon_3) S_t \mathrm{d}P_t, \end{eqnarray} $

无风险资产价格$ B_t $满足

$ \begin{eqnarray*} \left\{ \begin{aligned} &\mathrm{d}S_t = rS_t\mathrm{d}t, \\ &S_0 = 1. \end{aligned} \right. \end{eqnarray*} $

其中$ S_0 $为原生资产的初始价格, $ \mu(t, S_t) $表示风险资产的期望收益率, $ \varepsilon_i(0 < \varepsilon_i < 1, i = 1, 2, 3) $表示风险资产波动率的摄动参数, $ \mu(t, S_t) $和$ \sigma_i(t, S_t; \varepsilon_i)(0 < \varepsilon_i < 1, i = 1, 2, 3) $为时间$ t $和股票价格$ S_t $的一般函数, $ q $是股息分红率,且$ B_t^H $, $ B_t $, $ P_t $是相互独立的,作为原生资产的股票是连续支付股息红利的.如果用常数$ r $表示无风险资产的利率, $ \mathscr{P}(K, m, t) $表示欧式固定履约回望看跌期权的价格,用$ \mathscr{C}(K, M, t) $表示欧式固定履约回望看涨期权的价格,则由文献[25]可知$ \mathscr{P}(K, m, t) $和$ \mathscr{C}(K, M, t) $为如下两个随机偏微分方程初值问题的解

(2.6) $ \begin{array}{l} \left\{ \begin{aligned}&\mathfrak{L}\mathscr{P} = 0, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (t, S)\in(0, T)\times\mathbb{R}^+, \\ &\mathscr{P}(K, m, t) = (K-m_0^T)^+, (t, m)\in[0, T]\times\mathbb{R}^+, \end{aligned} \right. \end{array} $

(2.7) $ \begin{array}{l} \left\{ \begin{aligned} &\mathfrak{L}\mathscr{C} = 0, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; (t, S)\in(0, T)\times\mathbb{R}^+, \\ &\mathscr{C}(K, M, t) = (M_0^T-K)^+, (t, M)\in[0, T]\times\mathbb{R}^+, \end{aligned} \right. \end{array} $

其中$ \mathfrak{L} $为混合分数跳扩散B-S算子,定义为

$ \begin{eqnarray*} \mathfrak{L}V& = &\frac{\partial V}{\partial t}+\bigg( H\sigma_1^2t^{2H-1} +\dfrac{1}{2}\sigma_2^2+\dfrac{1}{2}\lambda\sigma_3^2\bigg)S^2\dfrac{\partial^2 V}{\partial S^2}+(r-q-\lambda\sigma_3)S\dfrac{\partial V}{\partial S} -(r+\lambda)V \nonumber \\&&+ \left\{ \begin{aligned} &\lambda\mathbb{E}[V(S(1+j_t), \min\{J, S(1+j_t)\}, t)], \mbox{回望看涨}, \\ &\lambda\mathbb{E}[V(S(1+j_t), \max\{J, S(1+j_t)\}, t)], \mbox{回望看跌}. \end{aligned}\right. \end{eqnarray*} $

定义如下符号

(3.1) $ \begin{eqnarray} \mathfrak{L}_0V& = &\partial_tV+\bigg[Ht^{2H-1}f(0, 0) +\dfrac{1}{2}g(0, 0)+\dfrac{1}{2}\lambda h(0, 0)\bigg](\partial_{xx}V-\partial_xV)+ \\&&+(r-q-\lambda\sqrt{h(0, 0)})(\partial_xV-V) -(r+\lambda)V \\&&+ \left\{ \begin{aligned} &\lambda\mathbb{E}[V(S(1+j_t), \min\{J, S(1+j_t)\}, t)], \mbox{回望看涨}, \\ &\lambda\mathbb{E}[V(S(1+j_t), \max\{J, S(1+j_t)\}, t)], \mbox{回望看跌}. \end{aligned}\right. \end{eqnarray} $

(3.2) $ \begin{eqnarray} \mathfrak{L}_{(\varepsilon_1, \varepsilon_2, \varepsilon_3)}V & = &\partial_tV+\bigg[Ht^{2H-1}f(t, x, \varepsilon_1) +\dfrac{1}{2}g(t, x, \varepsilon_2)+\dfrac{1}{2}\lambda h(t, x, \varepsilon_3)\bigg](\partial_{xx}V-\partial_xV) \\&&+(r-q-\lambda\sqrt{h(t, x, \varepsilon_3)})(\partial_xV-V) -(r+\lambda)V \\&&+ \left\{ \begin{aligned} &\lambda\mathbb{E}[V(S(1+j_t), \min\{J, S(1+j_t)\}, t)], \mbox{回望看涨}, \\ &\lambda\mathbb{E}[V(S(1+j_t), \max\{J, S(1+j_t)\}, t)], \mbox{回望看跌}. \end{aligned}\right. \end{eqnarray} $

其中

(3.3) $ \begin{equation} \begin{split} \left. \begin{aligned} &f(t, x, \varepsilon_1) = \sigma_1^2(t, x, \varepsilon_1), g(t, x, \varepsilon_2) = \sigma_2^2(t, x, \varepsilon_2), h(t, x, \varepsilon_3) = \sigma_3^2(t, x, \varepsilon_3), \\&x = \ln\bigg[S\prod\limits_{\mathbb{i} = 1}^n(1+j_{t_\mathbb{i}})\bigg].\end{aligned} \right\} \end{split} \end{equation} $

在如上变换下,当$ \varepsilon_i(0 < \varepsilon_i < 1, i = 1, 2, 3) = 0 $时, $ f(t, x, \varepsilon_1), g(t, x, \varepsilon_2), h(t, x, \varepsilon_3) $均为与$ t $和$ x $无关的常数,即$ f(t, x, 0) = f_0(0, 0), g(t, x, 0) = g_0(0, 0), h(t, x, 0) = h_0(0, 0) $,并且$ f(t, x, \varepsilon_1), g(t, x, \varepsilon_2), h(t, x, \varepsilon_3) $可以根据泰勒公式在$ \varepsilon_i(0 < \varepsilon_i < 1, i = 1, 2, 3) = 0 $附近展成如下形式的幂级数

(3.4) $ \begin{eqnarray} \left\{ \begin{aligned}&f(t, x, \varepsilon_1) = f_0(0, 0)+\varepsilon_1f_1(t, x)+\cdots+\frac{\varepsilon_1^n}{n!}f_n(t, x)+\cdots;\\ &g(t, x, \varepsilon_2) = g_0(0, 0)+\varepsilon_2g_1(t, x)+\cdots+\frac{\varepsilon_2^n}{n!}g_n(t, x)+\cdots;\\ &h(t, x, \varepsilon_3) = h_0(0, 0)+\varepsilon_3h_1(t, x)+\cdots+\frac{\varepsilon_3^n}{n!}h_n(t, x)+\cdots, \end{aligned} \right. \end{eqnarray} $

其中$ f_n(t, x), g_n(t, x), h_n(t, x) $分别表示$ f(t, x, \varepsilon_1), g(t, x, \varepsilon_2), h(t, x, \varepsilon_3) $对$ \varepsilon_i $$ (0 < \varepsilon_i < 1, $$ i = 1, 2, 3) $的$ n $阶导数在$ \varepsilon_i $$ (0 < \varepsilon_i < 1, i = 1, 2, 3) = 0 $处的值.由于$ f(t, x, \varepsilon_1) = \sigma_1^2(t, x, \varepsilon_1) > 0, $$ g(t, x, \varepsilon_2) = \sigma_2^2(t, x, \varepsilon_2) > 0, h(t, x, \varepsilon_3) = \sigma_3^2(t, x, \varepsilon_3) > 0, $于是这里恒假设

$ \begin{array}{l} f_0(0, 0)>0, g_0(0, 0)>0, h_0(0, 0)>0. \end{array} $

下面对随机偏微分方程初值问题(2.6)采用多尺度参数摄动方法作处理来解决欧式固定履约价的回望期权定价问题.令$ \mathscr{P}(K, m, t) = V(m, x, t) $,则初值问题(2.6)转化为

(3.5) $ \begin{equation} \left\{ \begin{array}{ll} \mathfrak{L}V = 0, &(t, m, x)\in(0, T)\times\mathbb{R}^+\times\mathbb{R}^+, \\ V(m, x, t) = (K-m_0^T)^+, & (t, m, x)\in[0, T]\times\mathbb{R}^+\times\mathbb{R}^+, \end{array} \right. \end{equation} $

根据多尺度参数摄动方法的思想^[16-19],把$ V(m, x, t) $在$ \varepsilon_i(0 < \varepsilon_i < 1, i = 1, 2, 3) = 0 $附近展开成幂级数

(3.6) $ \begin{eqnarray} V(m, x, t)& = &V_{0, 0, 0}(m, x, t)+\varepsilon_1V_{1, 0, 0}(m, x, t)+\varepsilon_2V_{0, 1, 0}(m, x, t) +\varepsilon_3V_{0, 0, 1}(m, x, t)+\cdots \\&&+\varepsilon_1^i\varepsilon_2^j\varepsilon_3^kV_{i, j, k}(m, x, t)+\cdots. \end{eqnarray} $

把上述级数(3.6)代入到随机偏微分方程初值问题(3.5),可得

$ \begin{eqnarray*} 0& = &{\mathfrak L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3}V \nonumber \\ & = &{\mathfrak L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3}V_{0, 0, 0} +\varepsilon_1{\mathfrak L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3}V_{1, 0, 0} +\varepsilon_2{\mathfrak L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3}V_{0, 1, 0} +\varepsilon_3{\mathfrak L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3}V_{0, 0, 1}+\cdots \\ &&+\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k {\mathfrak L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3}V_{i, j, k}+\cdots \nonumber \\ & = &{\mathfrak L}_0V_{0, 0, 0}+ \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_1^n}{n!}f_n(m, x, t)\bigg\} (\partial_{xx}V_{0, 0, 0}-\partial_xV_{0, 0, 0})\\ &&+\frac{1}{2} \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_2^n}{n!}g_n(m, x, t)\bigg\} (\partial_{xx}V_{0, 0, 0}-\partial_xV_{0, 0, 0})\\ &&+\frac{1}{2}\lambda \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_3^n}{n!}h_n(m, x, t)\bigg\} (\partial_{xx}V_{0, 0, 0}-\partial_xV_{0, 0, 0})-(r+\lambda)V_{0, 0, 0} +\lambda{\Bbb E}[V_{0, 0, 0}(\cdot)] \\ && +\varepsilon_1{\mathfrak L}_0V_{1, 0, 0}+ \varepsilon_1\bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_1^n}{n!}f_n(m, x, t)\bigg\} (\partial_{xx}V_{1, 0, 0}-\partial_xV_{1, 0, 0})\\ &&+\frac{1}{2}\varepsilon_1 \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_2^n}{n!}g_n(m, x, t)\bigg\}\nonumber (\partial_{xx}V_{1, 0, 0}-\partial_xV_{1, 0, 0})\\ &&+\frac{1}{2}\lambda\varepsilon_1 \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_3^n}{n!}h_n(m, x, t)\bigg\} (\partial_{xx}V_{1, 0, 0}-\partial_xV_{1, 0, 0})-(r+\lambda)V_{1, 0, 0} +\lambda{\Bbb E}[V_{1, 0, 0}(\cdot)] \nonumber \\&&+\varepsilon_2{\mathfrak L}_0V_{0, 1, 0}+ \varepsilon_2\bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_1^n}{n!}f_n(m, x, t)\bigg\} (\partial_{xx}V_{0, 1, 0}-\partial_xV_{0, 1, 0})\\ &&+\frac{1}{2}\varepsilon_2 \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_2^n}{n!}g_n(m, x, t)\bigg\}\nonumber (\partial_{xx}V_{0, 1, 0}-\partial_xV_{0, 1, 0})\\ &&+\frac{1}{2}\lambda\varepsilon_2 \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_3^n}{n!}h_n(m, x, t)\bigg\} (\partial_{xx}V_{0, 1, 0}-\partial_xV_{0, 1, 0})-(r+\lambda)V_{0, 1, 0} +\lambda{\Bbb E}[V_{0, 1, 0}(\cdot)]\\ &&+\varepsilon_3{\mathfrak L}_0V_{0, 0, 1}+ \varepsilon_3\bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_1^n}{n!}f_n(m, x, t)\bigg\} (\partial_{xx}V_{0, 0, 1}-\partial_xV_{0, 0, 1})\nonumber \\&&+\frac{1}{2}\varepsilon_3 \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_2^n}{n!}g_n(m, x, t)\bigg\} (\partial_{xx}V_{0, 0, 1}-\partial_xV_{0, 0, 1})\\ &&+\frac{1}{2}\lambda\varepsilon_3 \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_3^n}{n!}h_n(m, x, t)\bigg\} (\partial_{xx}V_{0, 0, 1}-\partial_xV_{0, 0, 1})-(r+\lambda)V_{0, 0, 1} +\lambda{\Bbb E}[V_{0, 0, 1}(\cdot)] +\cdots\\ &&+\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k{\mathfrak L}_0V_{i, j, k}+ \varepsilon_1^i\varepsilon_2^j\varepsilon_3^k \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_1^n}{n!}f_n(m, x, t)\bigg\} (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k})\\ &&+\frac{1}{2} \varepsilon_1^i\varepsilon_2^j\varepsilon_3^k \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_2^n}{n!}g_n(m, x, t)\bigg\} (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k})\\ &&+\frac{1}{2}\lambda \varepsilon_1^i\varepsilon_2^j\varepsilon_3^k \bigg\{\sum\limits_{n = 1}^\infty\frac{\varepsilon_3^n}{n!}h_n(m, x, t)\bigg\} \nonumber (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k})\\ &&-(r+\lambda)V_{i, j, k} +\lambda{\Bbb E}[V_{i, j, k}(\cdot)]+\cdots, \end{eqnarray*} $

其中$ {\Bbb E} $为二元可微函数$ V $的期望算子,比较$ \varepsilon_1, \varepsilon_2, \varepsilon_3 $的同次幂系数,可得

$ \begin{eqnarray*} 0& = &{\mathfrak L}_0V_{0, 0, 0} +\varepsilon_1\bigg\{{\mathfrak L}_0V_{1, 0, 0}+f_1(m, x, t) (\partial_{xx}V_{1, 0, 0}-\partial_xV_{1, 0, 0})\bigg\}-(r+\lambda)V_{1, 0, 0} \\ && +\lambda{\Bbb E}[V_{1, 0, 0}(\cdot)] +\varepsilon_2\bigg\{{\mathfrak L}_0V_{0, 1, 0}+\frac{1}{2}g_1(m, x, t) (\partial_{xx}V_{0, 1, 0}-\partial_xV_{0, 1, 0})\bigg\}-(r+\lambda)V_{0, 1, 0} \\ && +\lambda{\Bbb E}[V_{0, 1, 0}(\cdot)] +\varepsilon_3\bigg\{{\mathfrak L}_0V_{0, 0, 1}+\frac{1}{2}\lambda h_1(m, x, t) (\partial_{xx}V_{0, 0, 1}-\partial_xV_{0, 0, 1})\bigg\}-(r+\lambda)V_{0, 0, 1} \\ && +\lambda{\Bbb E}[V_{0, 0, 1}(\cdot)]+\cdots +\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k \bigg\{{\mathfrak L}_0V_{i, j, k}+\sum\limits_{n = 1}^\infty\frac{\varepsilon_1^n}{n!}f_n(m, x, t) (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k})\bigg\} \\ &&+\frac{1}{2}\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k \sum\limits_{n = 1}^\infty\frac{\varepsilon_2^n}{n!}g_n(m, x, t) (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k}) \\ &&+\frac{1}{2}\lambda\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k \sum\limits_{n = 1}^\infty\frac{\varepsilon_3^n}{n!}h_n(m, x, t) (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k})-(r+\lambda)V_{i, j, k} +\lambda{\Bbb E}[V_{i, j, k}(\cdot)]+\cdots. \end{eqnarray*} $

由$ x $和$ t $的任意性,令$ \varepsilon_1, \varepsilon_2, \varepsilon_3 $的相同次幂项的系数为零,从而可得

$ \begin{eqnarray*} \mathfrak{L}_0V_{0, 0, 0} = 0, \\ \mathfrak{L}_0V_{1, 0, 0}+f_1(m, x, t) (\partial_{xx}V_{1, 0, 0}-\partial_xV_{1, 0, 0}) = 0, \\ \mathfrak{L}_0V_{0, 1, 0}+\dfrac{1}{2}g_1(m, x, t) (\partial_{xx}V_{0, 1, 0}-\partial_xV_{0, 1, 0}) = 0, \\ \mathfrak{L}_0V_{0, 0, 1}+\dfrac{1}{2}\lambda h_1(m, x, t) (\partial_{xx}V_{0, 0, 1}-\partial_xV_{0, 0, 1}) = 0, \\ \mathfrak{L}_0V_{i, j, k}+\mathscr{U}_{i, j, k}(m, x, t) = 0, \end{eqnarray*} $

其中

$ \begin{eqnarray*} \mathscr{U}_{i, j, k}(m, x, t)& = &\sum\limits_{n = 1}^\infty\frac{\varepsilon_1^n}{n!}f_n(m, x, t) (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k})\\ &&+\frac{1}{2} \sum\limits_{n = 1}^\infty\frac{\varepsilon_2^n}{n!}g_n(m, x, t) (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k}) \\&&+ \frac{1}{2}\lambda \sum\limits_{n = 1}^\infty\frac{\varepsilon_3^n}{n!}h_n(m, x, t) (\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k})-(r+\lambda)V_{i, j, k} +\lambda\mathbb{E}[V_{i, j, k}(\cdot)]. \end{eqnarray*} $

注意到初值问题(3.5)的初值条件复杂,不妨根据幂级数展式(3.6)对初值问题(3.5)的初值条件进行分解,把抛物型混合跳扩散随机偏微分方程(3.5)的求解问题化归为一组可求解的常系数抛物型混合跳扩散随机偏微分方程的初值问题,其中$ V_{0, 0, 0}(m, x, t) $为如下抛物型混合跳扩散随机偏微分方程初值问题的解

(3.7) $ \begin{equation} \left\{ \begin{aligned}&\mathfrak{L}_0V_{0, 0, 0} = 0, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \ (t, m, x)\in(0, T)\times\mathbb{R}^+\times\mathbb{R}^+, \\ &V_{0, 0, 0}(m, x, T) = (K-m_0^T)^+, (t, m, x)\in[0, T]\times\mathbb{R}^+\times\mathbb{R}^+, \end{aligned} \right. \end{equation} $

$ V_{i, j, k}(m, x, t) $为如下抛物型混合跳扩散随机偏微分方程初值问题的解

(3.8) $ \begin{equation} \left\{ \begin{aligned}&\mathfrak{L}_0V_{i, j, k}+\mathscr{U}_{i, j, k}(m, x, t) = 0, (t, m, x)\in(0, T)\times\mathbb{R}^+\times\mathbb{R}^+, \\ &V_{i, j, k}(m, x, T) = 0, \; \; \; \; \; \; \; \; \; \; \; \; \; (t, m, x)\in[0, T]\times\mathbb{R}^+\times\mathbb{R}^+. \end{aligned} \right. \end{equation} $

下面依次计算$ V_{i, j, k} $,其中$ i = 0, 1, 2, \cdots; j = 0, 1, 2, \cdots; k = 0, 1, 2, \cdots $.

引理4.1  若$ V_{0, 0, 0}(m, x, t) $为抛物型混合跳扩散随机偏微分方程初值问题(3.7)的解,则有

$ \begin{eqnarray*} V_{0, 0, 0}(m, x, t)& = &\exp\{-r(T-t)\}(K-m) +\exp\{-r(T-t)\}K{\Bbb N}(-d_2)\\ &&-\exp\{[-q-\lambda\sqrt{h_0(0, 0)}](T-t)\} m {\Bbb N}(-d_1)\\ &&+\frac{r}{2[-q-\lambda\sqrt{h_0(0, 0)}]} \bigg[\exp\{[-q-\lambda\sqrt{h_0(0, 0)}](T-t)\}{\Bbb N}(d_1) \\&&-\exp\{-r(T-t)\}\exp\left\{\frac{2K[-q-\lambda\sqrt{h_0(0, 0)}]}{r}\right\} {\Bbb N}(d_1^r)\bigg], \end{eqnarray*} $

其中

$ \begin{eqnarray*} d_1& = &\bigg\{x-\ln K+[r-q-\lambda\sqrt{h_0(0, 0)}](T-t)+f_0(0, 0)(T^{2H}-t^{2H}) \\&&+\frac{1}{2}[g_0(0, 0)+\lambda h_0(0, 0)](T-t)\bigg\} \\&&\cdot\bigg[\sqrt{2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)}\bigg]^{-1}, \end{eqnarray*} $

$ \begin{eqnarray*} d_2& = &\bigg\{x-\ln K-[r-q-\lambda\sqrt{h_0(0, 0)}](T-t)+f_0(0, 0)(T^{2H}-t^{2H}) \\&&+\frac{1}{2}[g_0(0, 0)+\lambda h_0(0, 0)](T-t)\bigg\} \\&&\cdot\bigg[\sqrt{2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)}\bigg]^{-1}, \end{eqnarray*} $

$ \begin{eqnarray*} d_1^r = d_1-\frac{2[-q-\lambda\sqrt{h_0(0, 0)}]}{\sqrt{r}}\sqrt{T-t}, \end{eqnarray*} $

$ K $为固定履约敲定价格, $ {\Bbb N}(\cdot) $为累积正态分布函数.

证明详见文献[25].

引理4.2  线性抛物型混合跳扩散随机偏微分方程(3.8)的解可表示为

$ V_{i, j, k}(m, x, t) = \exp\{{\bf a}x+{\bf b}(T-t)\}\int_{-\infty}^{+\infty}\int_t^T \widetilde{{\mathscr U}}_{i, j, k}(m, x, T-s)G_0(\xi, x, s){\rm d}s{\rm d}\xi, $

其中

$ \widetilde{{\mathscr U}}_{i, j, k}(m, x, s) = \exp\{-{\bf a}x-{\bf b}s\}{\mathscr U}_{i, j, k}(m, x, T-s), $

$ \begin{eqnarray*} G_0(\xi, x, t) & = &\sum\limits_{n = 0}^\infty\left\{\frac{\lambda^n(T-t)^n\exp [ -\lambda(T-t)]}{n!}{\cal E}_n\right\}\\ &&\cdot\frac{1} {\xi\sqrt{2\pi[2f_0(0, 0)(T^{2H}-t^{2H})+g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)]}}\\ &&\cdot\exp{\Bigg\{-\frac{(x-\xi)^2}{2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)}}\Bigg\}, \end{eqnarray*} $

这里$ {\cal E}_n $表示$ \prod\limits_{i = 1}^n (1+j_{t_i}) $的期望算子,且

$ \begin{eqnarray*} &&{\bf a}(T, t) = -r(T-t)\nonumber\\ && -\frac{\Big[(r-q-\lambda \sqrt{h_0(0, 0)})(T-t)-f_0(0, 0)(T^{2H}-t^{2H}) -\frac{1}{2}[g_0(0, 0)+\lambda h_0(0, 0)](T-t) \Big]^2}{2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)}, \end{eqnarray*} $

$ \begin{eqnarray*} {\bf b}(T, t)& = &\frac{\sum\limits_{n = 0}^{\infty}P\{j_t = n\}{\mathcal E}_n}{\xi\sqrt{2\pi[2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)]}} \nonumber \\ & = &\frac{\sum\limits_{n = 0}^\infty \left\{\lambda^n(T-t)^n\exp [ -\lambda(T-t)](n!)^{-1}{\cal E}_n\right\} }{\xi\sqrt{2\pi[2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)]}}. \end{eqnarray*} $

证  由回望期权的定义,在有效回望期内的任意时刻的价格$ m $是已知的,于是在计算$ V_{i, j, k}(m, x, t) $时, $ {\mathscr U}_{i, j, k}(m, x, t) $是已知的,因此将$ m $视为常数,并作变换

(4.1) $ \begin{eqnarray} u_{i, j, k}(x, t) = \exp\{-{\bf a}x-{\bf b}(T-t)\}V_{i, j, k}(m, x, s), s = T-t, \end{eqnarray} $

则抛物型混合跳扩散随机偏微分方程初值问题(3.8)可以转换为如下形式

(4.2) $ \begin{equation} \left\{ \begin{array}{ll} \partial_su_n-\bigg[Ht^{2H-1}f(0, 0) +\frac{1}{2}g(0, 0)+\frac{1}{2}\lambda h(0, 0)\bigg]\partial_{xx}u_n-[r-q-\lambda\sqrt{h(0, 0)}]\partial_xu_n\\ +(r+\lambda)u_n -\lambda{\Bbb E}[u_n(\cdot)] = \widetilde{{\mathscr U}}_{i, j, k}(m, x, t), \; \; \; \; (s, x)\in(0, T)\times\mathbb{R} ^+, \\ u_n(0, x) = 0, \hskip 5.8cm x\in \mathbb{R} ^+. \end{array} \right. \end{equation} $

由于$ {\mathscr U}_{i, j, k}(m, x, t) $是解析的,且$ \widetilde{{\mathscr U}}_{i, j, k}(m, x, s) = \exp\{-{\bf a}x-{\bf b}s\}{\mathscr U}_{i, j, k}(m, x, T-s) $,根据文献[16-19],抛物型混合跳扩散随机偏微分方程初值问题(4.2)的解可用如下Volterra积分表示

(4.3) $ \begin{eqnarray} u_{i, j, k}(x, t) = \int_{-\infty}^{+\infty}\int_t^T \widetilde{{\mathscr U}}_{i, j, k}(m, x, T-s)G_0(\xi, x, s){\rm d}s{\rm d}\xi, \end{eqnarray} $

对变换(4.1)进行逆变换可完成引理的证明.

由引理4.1,引理4.2以及公式(3.5)和(3.6),易得如下结论成立.

定理4.1  设$ {\mathscr P}(K, m, t) $表示欧式固定履约回望看跌期权的价格,则有

(4.4) $ \begin{eqnarray} {\mathscr P}(K, m, t) & = &{\mathscr P}_{0, 0, 0}(K, m, t)+\varepsilon_1{\mathscr P}_{1, 0, 0}(K, m, t)+\varepsilon_2{\mathscr P}_{0, 1, 0}(K, m, t) +\varepsilon_3{\mathscr P}_{0, 0, 1}(K, m, t)+\cdots \\&&+\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k{\mathscr P}_{i, j, k}(K, m, t)+\cdots, \end{eqnarray} $

其中

$ \begin{eqnarray*} {\mathscr P}_{0, 0, 0}(K, m, t)& = &\exp\{-r(T-t)\}(K-m) +\exp\{-r(T-t)\}K{\Bbb N}(-d_2)\\ &&-\exp\{[-q-\lambda\sqrt{h_0(0, 0)}](T-t)\} m {\Bbb N}(-d_1)\\ &&+\frac{r}{2[-q-\lambda\sqrt{h_0(0, 0)}]} \bigg[\exp\{[-q-\lambda\sqrt{h_0(0, 0)}](T-t)\}{\Bbb N}(d_1) \\&&-\exp\{-r(T-t)\}\exp\left\{\frac{2K[-q-\lambda\sqrt{h_0(0, 0)}]}{r}\right\} {\Bbb N}(d_1^r)\bigg], \end{eqnarray*} $

$ {\mathscr P}_{i, j, k}(K, m, t) = V_{i, j, k}(m, x, t), $

这里$ d_1, d_2, d_1^r $见引理4.1.

同理可证明欧式固定履约回望看涨期权的定价公式,以如下定理给出.

定理4.2  设$ {\mathscr C}(K, M, t) $表示欧式固定履约回望看涨期权的价格,则有

(4.5) $ \begin{eqnarray} {\mathscr C}(K, M, t) & = &{\mathscr C}_{0, 0, 0}(K, M, t)+\varepsilon_1{\mathscr C}_{1, 0, 0}(K, M, t)+\varepsilon_2{\mathscr C}_{0, 1, 0}(K, M, t) +\varepsilon_3{\mathscr C}_{0, 0, 1}(K, M, t)+\cdots \\&&+\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k{\mathscr C}_{i, j, k}(K, M, t)+\cdots, \end{eqnarray} $

其中

$ \begin{eqnarray*} {\mathscr C}_{0, 0, 0}(K, M, t)& = &\exp\{-r(T-t)\}(M-K)+\exp\{[-q-\lambda\sqrt{h_0(0, 0)}](T-t)\}M{\Bbb N}(-d_1) \\ &&-\exp\{-r(T-t)\} K{\Bbb N}(-d_2) \\ &&+\frac{r}{2[-q-\lambda\sqrt{h_0(0, 0)}]} \bigg[\exp\{-r(T-t)\}\exp\left\{\frac{2K[-q-\lambda\sqrt{h_0(0, 0)}]}{r}\right\} {\Bbb N}(d_1^r) \\&&-\exp\{[-q-\lambda\sqrt{h_0(0, 0)}](T-t)\}{\Bbb N}(d_1)\bigg], \end{eqnarray*} $

$ {\mathscr C}_{i, j, k}(K, M, t) = V_{i, j, k}(M, x, t), $

这里$ d_1, d_2, d_1^r $见引理4.1.

本节内容讨论近似结论(4.4), (4.5)的误差估计问题.先作如下假设

(ⅰ)假设$ \sqrt{f(t, x;\varepsilon_1)} $, $ \sqrt{g(t, x;\varepsilon_2)} $, $ \sqrt{h(t, x;\varepsilon_3)} $均在$ [0, T]\times\mathbb{R}^+ $上满足关于$ x $的一致Lipschtiz条件:存在正常数$ \mathcal {M}_1 $, $ \mathcal {M}_2 $, $ \mathcal {M}_3 $,对任意的$ t\in[0, T] $以及$ x, y\in\mathbb{R}^+ $,有

$ \begin{array}{l} \begin{split} \left\{ \begin{aligned} \left|\sqrt{f(t, x;\varepsilon_1)}-\sqrt{f(t, y;\varepsilon_1)}\right|\leq \mathcal {M}_1|x-y|, \\ \left|\sqrt{g(t, x;\varepsilon_2)}-\sqrt{g(t, y;\varepsilon_2)}\right|\leq \mathcal {M}_2|x-y|, \\ \left|\sqrt{h(t, x;\varepsilon_3)}-\sqrt{h(t, y;\varepsilon_3)}\right|\leq \mathcal {M}_3|x-y|. \end{aligned} \right. \end{split} \end{array} $

(ⅱ)假设$ f_n(t, x) $, $ g_n(t, x) $, $ h_n(t, x) $均在$ [0, T]\times\mathbb{R}^+ $上一致有界,即对任意的$ (t, x)\in[0, T]\times\mathbb{R}^+ $,对任意的正整数$ n $,存在正常数$ \mathfrak{M}_1 $, $ \mathfrak{M}_2 $, $ \mathfrak{M}_3 $,使得

(5.1) $ \begin{equation} \left|f_n(t, x)\right|\leq\mathfrak{M}_1, \left|g_n(t, x)\right|\leq \mathfrak{M}_2, \left|h_n(t, x)\right|\leq \mathfrak{M}_3. \end{equation} $

(ⅲ)假设对任意的$ (t, x)\in[0, T]\times\mathbb{R}^+ $,有

$ \begin{array}{l} \begin{split} \left\{ \begin{aligned} \left|f(t, x;\varepsilon_1)-\sum\limits_{l = 0}^n\frac{\varepsilon_1^l}{l!}f_l(t, x)\right|\leq \frac{\mathfrak{M}_1}{(n+1)!}\varepsilon_1^{n+1}, \\ \left|g(t, x;\varepsilon_2)-\sum\limits_{l = 0}^n\frac{\varepsilon_2^l}{l!}g_l(t, x)\right|\leq \frac{\mathfrak{M}_2}{(n+1)!}\varepsilon_2^{n+1}, \\ \left|h(t, x;\varepsilon_3)-\sum\limits_{l = 0}^n\frac{\varepsilon_3^l}{l!}h_l(t, x)\right|\leq \frac{\mathfrak{M}_3}{(n+1)!}\varepsilon_3^{n+1}. \end{aligned} \right. \end{split} \end{array} $

很显然如上假设中$ f(t, x;\varepsilon_1) $, $ g(t, x;\varepsilon_2) $, $ h(t, x;\varepsilon_3) $以及$ \sqrt{f(t, x;\varepsilon_1)} $, $ \sqrt{g(t, x;\varepsilon_2)} $, $ \sqrt{h(t, x;\varepsilon_3)} $均在$ (t, x)\in[0, T]\times\mathbb{R}^+ $上关于$ x $满足线性增长条件.由公式(3.4)和公式(5.1)可得

$ \begin{eqnarray*} \left|f(t, x;\varepsilon_1)-f(t, y;\varepsilon_1)\right|& = \left(\sqrt{f(t, x;\varepsilon_1)}+\sqrt{f(t, y;\varepsilon_1)}\right)\cdot \left|\sqrt{f(t, x;\varepsilon_1)}-\sqrt{f(t, y;\varepsilon_1)}\right| \\&\leq 2\sqrt{\mathfrak{M}_1}\cdot\left|\sqrt{f(t, x;\varepsilon_1)} -\sqrt{f(t, y;\varepsilon_1)}\right|\leq 2\mathcal {M}_1\sqrt{\mathfrak{M}_1}\cdot|x-y|, \\ \left|g(t, x;\varepsilon_2)-g(t, y;\varepsilon_2)\right|& = \left(\sqrt{g(t, x;\varepsilon_2)}+\sqrt{g(t, y;\varepsilon_2)}\right)\cdot \left|\sqrt{g(t, x;\varepsilon_2)}-\sqrt{g(t, y;\varepsilon_2)}\right| \\&\leq 2\sqrt{\mathfrak{M}_2}\cdot\left|\sqrt{g(t, x;\varepsilon_2)} -\sqrt{g(t, y;\varepsilon_2)}\right|\leq 2\mathcal {M}_2\sqrt{\mathfrak{M}_2}\cdot|x-y|, \\ \left|h(t, x;\varepsilon_3)-h(t, y;\varepsilon_3)\right|& = \left(\sqrt{h(t, x;\varepsilon_3)}+\sqrt{h(t, y;\varepsilon_3)}\right)\cdot \left|\sqrt{h(t, x;\varepsilon_3)}-\sqrt{h(t, y;\varepsilon_3)}\right| \\&\leq 2\sqrt{\mathfrak{M}_3}\cdot\left|\sqrt{h(t, x;\varepsilon_3)} -\sqrt{h(t, y;\varepsilon_3)}\right|\leq 2\mathcal {M}_3\sqrt{\mathfrak{M}_3}\cdot|x-y|. \end{eqnarray*} $

为了方便证明,令$ \partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k} = \mathscr{V}_{i, j, k}(S, x, t) $,下面给出两个关于$ \mathscr{V}_{i, j, k}(S, x, t) $的必要引理.

引理5.1  对任意的$ (S, x, t)\in\mathbb{R}^+\times\mathbb{R}^+\times[0, T] $,存在不依赖$ S, x, t $的正常数$ \mathscr{M}_1, \mathscr{M}_2 $,使得

(5.2) $ \begin{equation} \left. \begin{aligned}& |\mathscr{V}_{0, 0, 0}(S, x, t)|\leq \dfrac{\mathscr{M}_1}{\sqrt{2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)}}, \\&|V_{0, 0, 0}(S, x, t)|\leq\mathscr{M}_2. \end{aligned} \right\} \end{equation} $

证  由引理4.1容易得到

(5.3) $ \begin{equation} \partial_xd_1 = \partial_xd_2 = -\bigg[2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)\bigg]^{-\frac{1}{2}}, \end{equation} $

以及

(5.4) $ \begin{eqnarray} \partial_xV_{0, 0, 0} & = &\exp\{-r(T-t)\}K\mathbb{N}'(-d_2)-\exp\{[-q-\lambda\sqrt{h_0(0, 0)}](T-t)\} S \mathbb{N}'(-d_1)\\ &&+\frac{r}{2[-q-\lambda\sqrt{h_0(0, 0)}]} \exp\{[-q-\lambda\sqrt{h_0(0, 0)}](T-t)\}\mathbb{N}'(d_1), \end{eqnarray} $

(5.5) $ \begin{equation} \partial_{xx}V_{0, 0, 0} = \bigg[2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)\bigg]^{-\frac{1}{2}}-\partial_xV_{0, 0, 0}, \end{equation} $

将公式(5.4)和(5.5)代入$ \mathscr{V}_{0, 0, 0}(S, x, t) $,对任意的$ \epsilon > 0 $,有

$ \begin{eqnarray*} |\partial_{xx}V_{i, j, k}-\partial_xV_{i, j, k}|\leq\bigg[2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)\bigg]^{-\frac{1}{2}}<\epsilon, \end{eqnarray*} $

于是$ \mathscr{V}_{0, 0, 0}(S, x, t) $在$ \mathbb{R}^+\times\mathbb{R}^+\times[0, T] $上有界,即存在正常数$ \mathscr{M}_0 $,使得

$ 0<\mathscr{V}_{0, 0, 0}(S, x, t)<\mathscr{M}_0, $

令$ \mathscr{M}_1 = \mathscr{M}_0\bigg[2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)\bigg]^{-\frac{1}{2}} $,则公式(5.2)中第一个不等式成立.下证公式(5.2)中第二个不等式.

根据引理4.2,抛物型混合跳扩散随机偏微分方程初值问题(3.7)的解$ V_{0, 0, 0}(m, x, t) $存在Green函数$ G(x, y, t) $,使得

(5.6) $ \begin{equation} V_{0, 0, 0}(x, t) = \exp\{-r(T-t)\}\int_{-\infty}^{+\infty}[K-m]^+G(x, y, t) \mathrm{d}y, \end{equation} $

由文献[16],存在正常数$ \mathscr{M}_3, \mathscr{M}_4 $,使得

(5.7) $ \begin{eqnarray} |G(x, y, t)|&\leq&\frac{\mathscr{M}_3}{\sqrt{2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)}}\\&&\cdot\exp\bigg\{\frac{-\mathscr{M}_4(x-y)^2} {2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)}\bigg\}. \end{eqnarray} $

将公式(5.7)代入公式(5.6)可得

$ |V_{0, 0, 0}(x, t)|\leq K\bigg|\int_{-\infty}^{+\infty}G(x, y, t) \mathrm{d}y\bigg|\leq K\mathscr{M}_3\left(\frac{\pi}{\mathscr{M}_4}\right)^{\frac{1}{2}}. $

于是式(5.2)中第二个不等式成立.

引理5.2  对任意的$ (S, x, t)\in\mathbb{R}^+\times\mathbb{R}^+\times[0, T] $和任意的非负整数$ i, j, k $,存在不依赖$ S, x, t $的正常数$ \mathscr{M}_1^*, \mathscr{M}_2^* $,使得

$ \begin{eqnarray*} &|\mathscr{V}_{i, j, k}(S, x, t)|\leq\mathscr{M}_1^*, \; \; |V_{i, j, k}(S, x, t)|\leq\mathscr{M}_2^*. \end{eqnarray*} $

证明类似引理5.1,结合基本假设(ⅱ)的公式(5.1)可得.

注意到公式(3.2)是初值问题(3.7)的抛物算子,于是可以考虑利用抛物方程的Feyman-Kac公式来研究$ V(S, x, t) $的近似解以及近似解的误差估计.

由以上的辅助引理,给出本节的两个主要结论.

定理5.1  对于充分小的$ T $,存在不依赖$ x, t $和$ S $的正常数$ \mathscr{M} $,对任意的$ \epsilon_1 > 0 $,有

$ |\mathscr{P}(S, m, t)-\mathscr{P}_{0, 0, 0}(S, m, t)|\leq\mathscr{M}\epsilon_1. $

证  令$ \mathcal {V}_{0, 0, 0}(x, m, t) = V_{0, 0, 0}(x, m, t)-V(x, m, t) $,由引理5.1可知$ \mathcal {V}_{0, 0, 0}(x, m, T) = 0 $,于是

(5.8) $ \begin{eqnarray} &&\mathfrak{L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3}\mathcal {V}_{0, 0, 0}(x, t) \\& = &\partial_t\mathcal {V}_{0, 0, 0} +\bigg\{\bigg[Ht^{2H-1}f_0(0, 0)+\dfrac{1}{2}g_0(0, 0)+\dfrac{1}{2}\lambda h_0(0, 0)\bigg] (\partial_{xx}V_{0, 0, 0}-\partial_xV_{0, 0, 0})\\&& +[r-q-\lambda\sqrt{h_0(0, 0)}](\partial_xV_{0, 0, 0}-V_{0, 0, 0}) \bigg\}-(r+\lambda)V_{0, 0, 0} +\lambda\mathbb{E}[V_{0, 0, 0}(\cdot)] \\ & = &\bigg\{Ht^{2H-1}[f(t, x;\varepsilon_1)-f_0(0, 0)]+\dfrac{1}{2} [g(t, x;\varepsilon_2)-g_0(0, 0)]+\dfrac{1}{2}\lambda[h(t, x;\varepsilon_3)-h_0(0, 0)]\bigg\} \\&&\cdot(\partial_{xx}V_{0, 0, 0}-\partial_xV_{0, 0, 0}) +[r-q-\lambda\sqrt{h_0(0, 0)}](\partial_xV_{0, 0, 0}-V_{0, 0, 0})-(r+\lambda)V_{0, 0, 0} \\&&+\lambda\mathbb{E}[V_{0, 0, 0}(\cdot)] \\& = &\bigg\{Ht^{2H-1}[f(t, x;\varepsilon_1)-f_0(0, 0)]+\dfrac{1}{2} [g(t, x;\varepsilon_2)-g_0(0, 0)]+\dfrac{1}{2}\lambda[h(t, x;\varepsilon_3)-h_0(0, 0)]\bigg\} \\&&\cdot\mathscr{V}_{0, 0, 0}(t, x) +[r-q-\lambda\sqrt{h_0(0, 0)}]\partial_xV_{0, 0, 0}+\mathfrak{L}_0V_{0, 0, 0}-(r+\lambda)V_{0, 0, 0} +\lambda\mathbb{E}[V_{0, 0, 0}(\cdot)] \\& = &\bigg\{Ht^{2H-1}[f(t, x;\varepsilon_1)-f_0(0, 0)]+\dfrac{1}{2} [g(t, x;\varepsilon_2)-g_0(0, 0)]+\dfrac{1}{2}\lambda[h(t, x;\varepsilon_3)-h_0(0, 0)]\bigg\} \\&&\cdot\mathscr{V}_{0, 0, 0}(t, x) +[r-q-\lambda\sqrt{h_0(0, 0)}]\partial_xV_{0, 0, 0}-(r+\lambda)V_{0, 0, 0} +\lambda\mathbb{E}[V_{0, 0, 0}(\cdot)]. \end{eqnarray} $

因此, $ \mathcal {V}_{0, 0, 0}(x, t) $为如下抛物型混合跳扩散随机偏微分方程初值问题的解

(5.9) $ \begin{equation} \left\{ \begin{aligned}&\mathfrak{L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3} \mathcal {V}_{0, 0, 0}(x, t) = \mathscr{H}_{0, 0, 0}(x, t), \; (t, x)\in(0, T)\times\mathbb{R}^+, \\ &\mathcal{V}_{0, 0, 0}(x, T) = 0, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \ x\in\mathbb{R}^+\times\mathbb{R}^+, \end{aligned} \right. \end{equation} $

其中

(5.10) $ \begin{eqnarray} \mathscr{H}_{0, 0, 0}(x, t) & = &\bigg\{Ht^{2H-1}[f(t, x;\varepsilon_1)-f_0(0, 0)]+\dfrac{1}{2} [g(t, x;\varepsilon_2)-g_0(0, 0)]\\ &&+\dfrac{1}{2}\lambda[h(t, x;\varepsilon_3) -h_0(0, 0)]\bigg\} \mathscr{V}_{0, 0, 0}(t, x)\\ &&+[r-q-\lambda\sqrt{h_0(0, 0)}]\partial_xV_{0, 0, 0}-(r+\lambda)V_{0, 0, 0} +\lambda\mathbb{E}[V_{0, 0, 0}(\cdot)]. \end{eqnarray} $

即

(5.11) $ \begin{equation} \left\{ \begin{aligned}&\partial_t\mathcal {V}_{0, 0, 0}+\bigg[Ht^{2H-1}f(t, x;\varepsilon_1) +\dfrac{1}{2}g(t, x;\varepsilon_2)+\dfrac{1}{2}\lambda h(t, x;\varepsilon_3)\bigg]\partial_{xx} \mathcal {V}_{0, 0, 0}\\ &+[r-q-\lambda\sqrt{h(t, x;\varepsilon_3)}]\partial_x\mathcal{V}_{0, 0, 0} -(r+\lambda)V_{0, 0, 0} +\lambda\mathbb{E}[V_{0, 0, 0}(\cdot)]\\ & = \mathscr{H}_{0, 0, 0}(x, t), (t, x)\in(0, T)\times\mathbb{R}^+, \\ &\mathcal{V}_{0, 0, 0}(x, T) = 0, \; \; \; \; (t, x)\in[0, T]\times\mathbb{R}^+, \end{aligned} \right. \end{equation} $

由引理5.1,引理5.2以及公式(5.1),对任意的$ \mathscr{E}_1 > 0 $, $ \mathscr{E}_2 > 0 $, $ \mathscr{E}_3 > 0 $, $ (t, x)\in[0, T]\times\mathbb{R}^+ $,有

(5.12) $ \begin{equation} \left. \begin{aligned} &\left|f(t, x;\varepsilon_1)-f_0(0, 0)\right|\leq\mathfrak{M}_1\mathscr{E}_1, \\& \left|g(t, x;\varepsilon_2)-g_0(0, 0)\right|\leq\mathfrak{M}_2\mathscr{E}_2, \\& \left|h(t, x;\varepsilon_3)-h_0(0, 0)\right|\leq\mathfrak{M}_3\mathscr{E}_3.\end{aligned} \right\} \end{equation} $

把公式(5.2)和公式(5.12)代入公式(5.11),对任意的$ (t, x)\in[0, T]\times\mathbb{R}^+ $可得

(5.13) $ \begin{eqnarray} |\mathscr{H}_{0, 0, 0}(t, x)|&\leq&\bigg(\mathfrak{M}_1\mathscr{E}_1 +\frac{1}{2}\mathfrak{M}_2\mathscr{E}_2+\frac{1}{2}\mathfrak{M}_3\mathscr{E}_3\bigg) \\&&\cdot \dfrac{\mathscr{M}_1}{\sqrt{2f_0(0, 0)(T^{2H}-t^{2H}) +g_0(0, 0)(T-t)+\lambda h_0(0, 0)(T-t)}} +\mathscr{M}_2. \end{eqnarray} $

不妨对抛物初值问题(5.11)的解$ \mathcal {V}_{0, 0, 0}(x, t) $使用Feynman-Kac公式作处理,则$ \mathcal {V}_{0, 0, 0}(x, t) $可表示为如下形式的条件概率

(5.14) $ \begin{equation} \mathcal {V}_{0, 0, 0}(t, x) = E\bigg[\int_t^T\exp\bigg\{-r(T-t)\bigg\} \mathscr{H}_{0, 0, 0}(\tau, S_\tau)\mathrm{d}\tau\bigg|\mathfrak{F}_t\bigg], \end{equation} $

其中$ S_t $为混合跳扩散随机微分方程

(5.15) $ \begin{equation} \mathrm{d}S_t = \mu S_t\mathrm{d}t+\sqrt{f(t, S_t;\varepsilon_1)} S_t\mathrm{d}B_t^H +\sqrt{g(t, S_t;\varepsilon_2)} S_t \mathrm{d}B_t+\sqrt{h(t, S_t;\varepsilon_3)} S_t \mathrm{d}P_t \end{equation} $

的解,由于随机微分方程(5.15)存在解$ \mathcal {V}_{0, 0, 0}(t, x) $,把公式(5.13)代入公式(5.14)可得$ \mathcal {V}_{0, 0, 0}(t, x) $的估计式

(5.16) $ \begin{eqnarray} |\mathcal {V}_{0, 0, 0}(t, x)|&\leq& E\bigg[\int_t^T| \mathscr{H}_{0, 0, 0}(\tau, S_\tau)|\mathrm{d}\tau\bigg|\mathfrak{F}_t\bigg] \\&\leq&\bigg(\mathfrak{M}_1\mathscr{E}_1 +\frac{1}{2}\mathfrak{M}_2\mathscr{E}_2+\frac{1}{2}\mathfrak{M}_3\mathscr{E}_3\bigg) \\&&\cdot \int_0^T\dfrac{\mathscr{M}_1}{\sqrt{2f_0(0, 0)(T^{2H}-\tau^{2H}) +g_0(0, 0)(T-\tau)+\lambda h_0(0, 0)(T-\tau)}}\mathrm{d}\tau\\ &&+\mathscr{M}_2, \end{eqnarray} $

显然瑕积分$ \int_0^T\frac{1}{\sqrt{2f_0(0, 0)(T^{2H}-\tau^{2H}) +g_0(0, 0)(T-\tau)+\lambda h_0(0, 0)(T-\tau)}}\mathrm{d}\tau $是收敛的,定理得证.

定理5.2  对于充分小的$ T $,使得$ \sum\limits_{i = 0}^n\sum\limits_{j = 0}^n\sum\limits_{k = 0}^n\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k\mathscr{P}(S, m, t) $在$ \mathbb{R}^+\times\mathbb{R}^+\times[0, T] $上一致收敛到$ \mathscr{P}(S, m, t) $,即对任意的正整数$ n $以及$ \epsilon > 0 $,存在不依赖$ x, t $的正常数$ \mathscr{M}^* $,使得

$ \begin{eqnarray*} \bigg|\mathscr{P}(S, m, t)-\sum\limits_{i = 0}^n\sum\limits_{j = 0}^n\sum\limits_{k = 0}^n \varepsilon_1^i\varepsilon_2^j\varepsilon_3^k\mathscr{P}(S, m, t)\bigg| \leq\mathscr{M}^*(\varepsilon_1^{n+1}+\varepsilon_2^{n+1}+\varepsilon_3^{n+1})<\epsilon. \end{eqnarray*} $

证  仿照定理5.1的证明,考察

$ \begin{eqnarray*} \mathcal {V}_{i, j, k}(x, m, t) = V_{i, j, k}(x, m, t)-V(x, m, t), \end{eqnarray*} $

其中

$ \begin{eqnarray*} V_{i, j, k}(x, m, t) = \sum\limits_{i = 0}^n\sum\limits_{j = 0}^n\sum\limits_{k = 0}^n \varepsilon_1^i\varepsilon_2^j\varepsilon_3^k\mathscr{P}(S, m, t). \end{eqnarray*} $

注意到$ \mathcal {V}_{i, j, k}(x, m, t) $满足初值条件$ \mathcal {V}_n(x, m, T) = 0 $,又因为

$ \begin{eqnarray*} \left\{ \begin{aligned}&\mathfrak{L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3} V_{0, 0, 0}(x, m, t) = 0, \; (x, m, t)\in\mathbb{R}^+\times\mathbb{R}^+\times(0, T), \\ &\mathfrak{L}_{0, 0, 0} V_{i, j, k}(x, m, t) = \sum\limits_{i = 0}^n\sum\limits_{j = 0}^n\sum\limits_{k = 0}^n \varepsilon_1^i\varepsilon_2^j\varepsilon_3^k\mathfrak{L}_{0, 0, 0}V_{i, j, k} \\ & = -\sum\limits_{i = 0}^n\sum\limits_{j = 0}^n\sum\limits_{k = 0}^n \varepsilon_1^i\varepsilon_2^j\varepsilon_3^kV_{i, j, k}^*, \ (x, m, t)\in\mathbb{R}^+\times\mathbb{R}^+\times[0, T], \end{aligned} \right. \end{eqnarray*} $

所以有$ \mathfrak{L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3}\mathcal {V}_{i, j, k}(x, m, t) = \mathscr{H}_{i, j, k}(x, m, t) $,其中

(5.17) $ \begin{eqnarray} \mathscr{H}_{i, j, k}(x, m, t) & = &\bigg\{Ht^{2H-1}\bigg[f(t, x;\varepsilon_1)- \sum\limits_{l = 0}^{n-1}\frac{\varepsilon_1^l}{l!}f_l(t, x)\bigg]+\dfrac{1}{2} \bigg[g(t, x;\varepsilon_2)- \sum\limits_{l = 0}^{n-1}\frac{\varepsilon_2^l}{l!}g_l(t, x)\bigg] \\ &&+\dfrac{1}{2}\lambda\bigg[h(t, x;\varepsilon_3)- \sum\limits_{l = 0}^{n-1}\frac{\varepsilon_3^l}{l!}h_l(t, x)\bigg]\bigg\} \cdot \sum\limits_{i = 0}^{n-1}\varepsilon_1^i\sum\limits_{j = 0}^{n-1}\varepsilon_2^j \sum\limits_{k = 0}^{n-1}\varepsilon_3^k\mathscr{V}_{i, j, k}(x, m, t) \\&& +\left[r-q-\lambda\sqrt{h(t, x;\varepsilon_3)- \sum\limits_{l = 0}^{n-1}\frac{\varepsilon_3^l}{l!}h_l(t, x)}\ \right ] \\&& \cdot\sum\limits_{i = 0}^{n-1}\varepsilon_1^i\sum\limits_{j = 0}^{n-1}\varepsilon_2^j \sum\limits_{k = 0}^{n-1}\varepsilon_3^k\mathscr{V}_{i, j, k}(x, m, t) -(r+\lambda)V_{i, j, k} +\lambda\mathbb{E}[V_{i, j, k}(\cdot)]. \end{eqnarray} $

于是$ \mathcal {V}_n(x, t) $为如下抛物型混合跳扩散随机偏微分方程初值问题的解

(5.18) $ \begin{equation} \left\{ \begin{aligned}&\mathfrak{L}_{\varepsilon_1, \varepsilon_2, \varepsilon_3} \mathcal {V}_{i, j, k}(x, t) = \mathscr{H}_{i, j, k}(x, t), \; (t, x)\in(0, T)\times\mathbb{R}^+, \\ &\mathcal{V}_{i, j, k}(x, T) = 0, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\in\mathbb{R}^+\times\mathbb{R}^+, \end{aligned} \right. \end{equation} $

为了估计$ \mathscr{H}_{i, j, k}(x, t) $,利用公式(5.1),把基本假设(ⅲ)代入公式(5.17)可得

(5.19) $ \begin{eqnarray} |\mathscr{H}_{i, j, k}(x, m, t)| &\leq&\bigg\{Ht^{2H-1}\frac{\mathfrak{M}_1}{(n-l+1)!}\varepsilon_1^{n-l+1}+\dfrac{1}{2} \frac{\mathfrak{M}_2}{(n-l+1)!}\varepsilon_2^{n-l+1} \\ &&+\dfrac{1}{2}\lambda\frac{\mathfrak{M}_3}{(n-l+1)!}\varepsilon_3^{n-l+1}\bigg\} \cdot\sum\limits_{i = 0}^{n-1}\varepsilon_1^i\sum\limits_{j = 0}^{n-1}\varepsilon_2^j \sum\limits_{k = 0}^{n-1}\varepsilon_3^k|\mathscr{V}_{i, j, k}(x, m, t)| \\ &&+\left[r-q-\lambda\sqrt{\frac{\mathfrak{M}_3}{(n-l+1)!}\varepsilon_3^{n-l+1}}\right ]\cdot\sum\limits_{i = 0}^{n-1}\varepsilon_1^i\sum\limits_{j = 0}^{n-1}\varepsilon_2^j \sum\limits_{k = 0}^{n-1}\varepsilon_3^k|\mathscr{V}_{i, j, k}(x, m, t)| \\ &&-|(r+\lambda)V_{i, j, k}| +|\lambda\mathbb{E}[V_{i, j, k}(\cdot)]|. \end{eqnarray} $

进一步将引理5.1的公式(5.2)以及引理5.2的结果代入到公式(5.19),对任意的$ \epsilon^* > 0 $,可得

(5.20) $ \begin{eqnarray} |\mathscr{H}_{i, j, k}(x, m, t)| &\leq&\bigg\{Ht^{2H-1}\frac{\mathfrak{M}_1}{(n-l+1)!}\varepsilon_1^{n-l+1}+\dfrac{1}{2} \frac{\mathfrak{M}_2}{(n-l+1)!}\varepsilon_2^{n-l+1} \\&&+\dfrac{1}{2}\lambda\frac{\mathfrak{M}_3}{(n-l+1)!}\varepsilon_3^{n-l+1}\bigg\} \cdot\sum\limits_{i = 0}^{n-1}\varepsilon_1^i\sum\limits_{j = 0}^{n-1}\varepsilon_2^j \sum\limits_{k = 0}^{n-1}\varepsilon_3^k\mathscr{M}_1^* \\&&+\left[r-q-\lambda\sqrt{\frac{\mathfrak{M}_3}{(n-l+1)!}\varepsilon_3^{n-l+1}}\right ]\cdot\sum\limits_{i = 0}^{n-1}\varepsilon_1^i\sum\limits_{j = 0}^{n-1}\varepsilon_2^j \sum\limits_{k = 0}^{n-1}\varepsilon_3^k\mathscr{M}_1^* \\&&-(r+\lambda)\mathscr{M}_2^* +\lambda\mathbb{E}[\mathscr{M}_2^*]<\epsilon^*. \end{eqnarray} $

由Feynman-Kac公式可知,抛物型混合随机初值问题(5.18)的解可表示为如下形式的条件概率问题

(5.21) $ \begin{equation} \mathcal {V}_{i, j, k}(t, x) = E\bigg[\int_t^T\exp\bigg\{-r(T-t)\bigg\} \mathscr{H}_{i, j, k}(\tau, S_\tau)\mathrm{d}\tau\bigg|\mathfrak{F}_t\bigg], \end{equation} $

其中$ S_t $满足混合跳扩散随机微分方程(5.15).将估计式(5.20)代入公式(5.21),对任意的$ \epsilon > 0 $,可得$ \mathcal {V}_{i, j, k} $的估计式

$ \begin{eqnarray*} |\mathcal {V}_{i, j, k}(t, x)|\leq E\bigg[\int_t^T\exp\bigg\{-r(T-t)\bigg\} |\mathscr{H}_{i, j, k}(\tau, S_\tau)|\mathrm{d}\tau\bigg|\mathfrak{F}_t\bigg] <\epsilon. \end{eqnarray*} $

证毕.

本节内容对上节给出的误差估计结果进行数值模拟分析.根据混合分数跳扩散Black-Scholes公式,当波动率$ \varepsilon_i(0 < \varepsilon_i < 1, i = 1, 2, 3) $为常数时,欧式固定履约价的回望期权是有精确解的,给定$ \sigma_1 = 0.1, \sigma_2 = 0.02, \sigma_3 = 0.0007 $,设定小参数$ \varepsilon_i(0 < \varepsilon_i < 1, i = 1, 2, 3) $使得$ \sigma_i(0 < \sigma_i < 1, i = 1, 2, 3) = \varepsilon_i+\sigma_i(0) $,根据定理1从$ \sigma_i(0) $开始依次模拟欧式固定履约价的回望看跌期权价格,然后与$ \sigma_1 = 0.1, \sigma_2 = 0.02, \sigma_3 = 0.0007 $时的期权价格的精确结果作比对.取无风险利率$ r = 0.025 $,提前约定的固定敲定价格$ K = 100 $, $ \lambda = 2.68, q = 0.01, H = 0.76 $, $ T = 1 $(年),则$ \mathscr{P}_i $的Matlab结果如下.

上述模拟中$ \mathscr{P}_n $表示为

$ \begin{eqnarray*} &&\mathscr{P}_n = \mathscr{P}_{0, 0, 0}(S, m, t)+\varepsilon_1\mathscr{P}_{1, 0, 0}(S, m, t)+\varepsilon_2\mathscr{P}_{0, 1, 0}(S, m, t) +\varepsilon_3\mathscr{P}_{0, 0, 1}(S, m, t)+\cdots \nonumber \\&&\; \; \; \; \; \; \; \; +\varepsilon_1^i\varepsilon_2^j\varepsilon_3^k\mathscr{P}_{i, j, k}(S, m, t). \end{eqnarray*} $

观察表 1和表 2可以看出随着阶数$ i, j, k $的增大,期权价格的近似解逐步逼近期权价格的精确解.对比图 1也可以看出当$ \varepsilon_1 = 0.01, \varepsilon_2 = 0.001, \varepsilon_3 = 0.0001 $时需要计算到$ \mathscr{P}_5 $才有较好的近似结果,这充分说明当小参数$ \varepsilon_i(0 < \varepsilon_i < 1, i = 1, 2, 3) $越小时期权价格逼近精确解的速度越快,这与定理1所得误差估计结论相符.