﻿ 广义自回归条件异方差模型加速模拟定价理论
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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (3): 435-443.  DOI: 10.11908/j.issn.0253-374x.2019.03.019 0

### 引用本文

MA Junmei, ZHUO Jinwu, ZHANG Jian, CHEN Lu. Pricing Accelerated Simulation Theory of Generalized Autoregressive Conditional Heteroskedasticity Model[J]. Journal of Tongji University (Natural Science), 2019, 47(3): 435-443. DOI: 10.11908/j.issn.0253-374x.2019.03.019

### 文章历史

1. 上海财经大学 数学学院，上海 200433;
2. 上海市金融信息技术研究重点实验室，上海 200433;
3. 应用数学福建省高校重点实验室(莆田学院)，福建 莆田 351100;
4. 上海财经大学 信息管理与工程学院，上海 200433

Pricing Accelerated Simulation Theory of Generalized Autoregressive Conditional Heteroskedasticity Model
MA Junmei 1,2,3, ZHUO Jinwu 4, ZHANG Jian 1, CHEN Lu 1
1. School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China;
2. Shanghai Key Laboratory of Financial Information Technology, Shanghai 200433, China;
3. Key Laboratory of Applied Mathematics, Fujian Province University (Putian University), Putian 351100, China;
4. School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai 200433, China
Abstract: The accelerated simulation pricing theory of variance derivatives under generalized auto regressive conditional heteroskedasticity (GARCH) stochastic volatility model was studied. Based on the analytical solution under the Black-Scholes model and their moments analysis of these two kinds of processes, a more efficient acceleration technique of control variate was proposed and the method of selecting optimal control variate was also given. The numerical results show that the proposed accelerated simulation method of control variate effectively reduce the simulation error and improve the computational efficiency. The algorithm can also be used to solve the computational problems of other complex products under GARCH stochastic volatility model, such as Asian option, Basket option, Capped variance swap, Corridor variance swap and Gamma variance swap, etc.
Key words: GARCH    stochastic volatility    accelerate    control variate    variance derivatives

GARCH模型是将波动率视为过去信息集的函数，是Bollerslev[3]在Engle[4]在ARCH模型基础上发展起来的.GARCH模型考虑了扰动项的滞后值和扰动项条件方差的滞后值，克服了ARCH模型无法反映波动率的持续性以及不能保证参数非负的缺点，被广泛用于描述金融市场上资产收益的波动变化.1987年，Bollerslev在考虑分布有偏和放宽假设的前提下，使用t分布代替正态分布描述时间序列的偏度和峰度问题，建立了AGARCH模型[4].基于这些研究，考虑到该模型的应用性和扩展性，此后20多年，许多学者针对不同问题构建出多种变形模型，构成GARCH族模型，使条件异方差结构得到完善的诠释.例如，Robert等为解决高风险高收益问题，将风险加入收益方程，建立了MGARCH模型[5].针对冲击非对称效应，Nelson提出了指数模型EGARCH[6], Zakoian提出了门限模型TGARCH[7].大量实证证明GARCH模型对金融时间序列有较好的描述，它充分体现了金融数据的特征，这些GARCH族模型的应用与实证领域双向作用，使得模型在分析波动性和表征宏观、金融高频数据等方面体现出巨大的价值和意义.

GARCH模型描述的是离散时间经济情形，在进行产品定价研究时，不能象连续时间经济情形那样，可以建立期权价格满足的偏微分方程，所以，Monte Carlo模拟方法被普遍用来解决GARCH模型下金融产品的计算问题.Duan等[8]在1995年第一次提出了一个在GARCH模型框架下欧式期权定价的完整理论，运用均衡定价原理证明了当投资者的效用函数满足一定的条件时，就存在一个满足局部风险中性定价关系的概率测度Q，用传统的Monte Carlo模拟算法计算了GARCH模型欧式期权的定价问题[8-9]；邵斌和丁娟研究了GARCH模型下美式亚式期权价值的Monte Carlo模拟算法，在Longstaff等的美式期权定价的最小二乘算法的基础上，开发了GARCH模型下美式亚式期权定价的最小二乘模拟算法[10].

 $P = M \cdot \left( {{\sigma ^2} - {K^2}} \right)$

Hull-White随机波动率模型下波动率衍生产品的定价问题.

1 Monte Carlo方法及控制变量技术

Monte Carlo方法是以概率论为基础，通过计算机生成符合一定概率分布随机数的一种数值计算方法.当要求解的问题是某一事件的概率，或某个随机变量的数学期望时，可以利用Monte Carlo方法求解，假设要估计随机变量V的期望E(V)，利用Monte Carlo法，根据随机变量V的概率分布得到n次重复抽样V1, V2, …, Vn，它们独立同分布且有有限期望值E(V)和有限方差值σ2, V1, V2, …, Vn的算术平均值V，它是E(V)的无偏估计值，即$\bar V = \frac{1}{n}\sum\limits_{j = 1}^n {{V_j}}$, 且E[V]=E[V]，Var[V]=$\frac{{{\sigma ^2}}}{n}$.由强大数定理可知，当样本数n充分大时，估计值V以概率1收敛到目标值E[V].

 $P\left( {\mathop {\lim }\limits_{n \to \infty } \bar V = E\left[ V \right]} \right) = 1$

 $P\left( {\left| {\bar V - E\left[ V \right]} \right| < \frac{{{\lambda _\alpha }\sigma }}{{\sqrt n }}} \right) \approx \mathit{\Phi }\left( {{\lambda _\alpha }} \right) = 1 - \alpha$

 ${V_j}\left( b \right) = {V_j} - b\left[ {{X_j} - E\left( X \right)} \right]$

 $\bar V\left( b \right) = \frac{1}{m}\sum\limits_{j = 1}^m {\left( {{V_i} - b\left( {{X_i} - E\left( X \right)} \right)} \right)}$

 $\begin{array}{*{20}{c}} {{\rm{Var}}\left( {{V_j}\left( b \right)} \right) = {\rm{Var}}\left( V \right) + {b^2}{\rm{Var}}\left( X \right) - }\\ {2b\sqrt {{\rm{Var}}\left( V \right)} \sqrt {{\rm{Var}}\left( X \right)} {\rho _{XV}}} \end{array}$

 $\frac{{{\rm{Var}}\left( {{V_i}\left( {{b^ * }} \right)} \right)}}{{{\rm{Var}}\left( {{V_i}} \right)}} = 1 - \rho _{XV}^2$

2 GARCH模型下方差衍生产品的控制变量加速模拟理论 2.1 GARCH随机波动率模型介绍

 $\ln \frac{{{S_{t + 1}}}}{{{S_t}}} = r - \frac{1}{2}\sigma _{t + 1}^2 + {\sigma _{t + 1}}{\varepsilon _{t + 1}}$ (1)

 $\sigma _t^2 = {\alpha _0} + {\alpha _1}{\left( {{\varepsilon _{t - 1}} - \left( {\lambda + \gamma } \right)} \right)^2}\sigma _{t - 1}^2 + {\beta _1}\sigma _{t - 1}^2$

 $\begin{array}{*{20}{c}} {V\left( {{S_T}, T} \right) = M \times \left( {\sum\limits_{i = 1}^N {{{\left( {\ln \frac{{{S_i}}}{{{S_{i - 1}}}}} \right)}^2}} - K_{{\rm{strike}}}^2} \right) \buildrel \Delta \over = }\\ {h\left( {{S_0}, {S_1}, \cdots , {S_N}} \right)} \end{array}$

 $V\left( {{S_0}, 0} \right) = E\left[ {{{\rm{e}}^{ - rT}}h\left( {{S_0}, {S_1}, \cdots , {S_N}} \right)} \right]$

2.2 Black-Scholes框架下方差互换产品价格的解析解

 $\frac{{{\rm{d}}{S_{\rm{t}}}}}{{{S_{\rm{t}}}}} = r{\rm{d}}t + {\sigma _{\rm{c}}}{\rm{d}}{B_{\rm{t}}}$ (2)

 ${S_{{T_i}}} = {S_{{T_{i - 1}}}}\exp \left( {r - \frac{1}{2}\sigma _{\rm{c}}^2} \right)\Delta {T_i} + {\sigma _{\rm{c}}}\sqrt {\Delta {T_i}} {X_i}$

 $\begin{array}{l} h\left( {{S_0}, {S_1}, \cdots , {S_N}} \right) = h\left( {{S_0}, {S_0}{{\rm{e}}^{\left( {r - \frac{1}{2}\sigma _{\rm{c}}^2} \right){T_1} + {\sigma _{\rm{c}}}\sqrt {{T_1}} {X_1}}}, } \right.\\ \left. { \cdots , {S_0}{{\rm{e}}^{\left( {r - \frac{1}{2}\sigma _{\rm{c}}^2} \right){T_N} + {\sigma _{\rm{c}}}\sqrt {{T_N}} {X_N}}}} \right) \end{array}$

 $\begin{array}{l} W\left( {{S_0}, 0} \right) = E\left[ {{e^{ - rT}}h\left( {{S_0}, {S_1}, \cdots , {S_N}} \right)} \right] = \\ \int { \cdots \int {h\left( {{S_0}, {S_0}{{\rm{e}}^{\left( {r - \frac{1}{2}\sigma _{\rm{c}}^2} \right){T_1} + {\sigma _{\rm{c}}}\sqrt {{T_1}} {X_1}}}, \cdots , } \right.} } \\ \left. {{S_0}{{\rm{e}}^{\left( {r - \frac{1}{2}\sigma _{\rm{c}}^2} \right){T_N} + {\sigma _{\rm{c}}}\sqrt {{T_N}} {X_N}}}} \right) \cdot \\ {\left( {\frac{1}{{\sqrt {2{\rm{ \mathit{ π} }}} }}} \right)^N}{{\rm{e}}^{ - \frac{{x_1^2 + x_2^2 + \cdots + x_N^2}}{2}}}{\rm{d}}{x_1}{\rm{d}}{x_2} \cdots {\rm{d}}{x_N} \end{array}$

 $\begin{array}{l} W\left( {{S_0}, 0} \right) = {{\rm{e}}^{ - rT}}M\left[ {\sigma _{\rm{c}}^2{\rm{T}} + \left( {{r^2} - r\sigma _{\rm{c}}^2 + \frac{1}{4}\sigma _{\rm{c}}^4} \right)\sum\limits_{i = 1}^N {\Delta t_i^2} - } \right.\\ \left. {K_{{\rm{Var}}}^2} \right]. \end{array}$
2.3 GARCH波动率模型产品定价的控制变量加速模拟算法

(1) 步骤1。n等分时间段[0, T]，等分间隔$\Delta t = \frac{T}{n}$，把每一个间隔都作为观察点，基于常数波动率的收益率变化过程有式(3)：

 $\left\{ \begin{array}{l} {S_{\left( 1 \right)}}\left( {{t_{i + 1}}} \right) = {S_{\left( 1 \right)}}\left( {{t_i}} \right){{\rm{e}}^{\left( {r - \frac{1}{2}\sigma _{\rm{c}}^2} \right)\Delta t + {\sigma _{\rm{c}}}\sqrt {{\Delta _t}} {\varepsilon _{1, i}}}}\\ {S_{\left( 1 \right)}}\left( {{t_0}} \right) = {S_0} \end{array} \right.$ (3)

(2) 步骤2.运用方差互换产品定价模型和步骤1中模拟路径j, 可求出控制变量在零时刻的价格为

 ${X_j} = M{{\rm{e}}^{ - rT}}\left[ {\frac{1}{T}\sum\limits_{i = 1}^N {{{\left( {\ln \frac{{{S_{1, j}}\left( {{t_i}} \right)}}{{{S_{1, j}}\left( {{t_{i - 1}}} \right)}}} \right)}^2}} - K_{{\rm{Var}}}^2} \right]$ (4)

(3) 步骤3.n等分时间段[0, T]，基于GARCH模型的收益率变化过程(在于步骤1中相同的随机序列ε1下生成)，有式(5)：

 $\left\{ \begin{array}{l} {S_{\left( 2 \right)}}\left( {{t_{i + 1}}} \right) = {S_{\left( 2 \right)}}\left( {{t_i}} \right){{\rm{e}}^{\left( {r - \frac{1}{2}{\sigma ^2}\left( {{t_i}} \right)} \right)\Delta t + \sigma \left( {{t_i}} \right)\sqrt {\Delta t} {\varepsilon _{1, i}}}}\\ {S_{\left( 2 \right)}}\left( {{t_0}} \right) = {S_0} \end{array} \right.$ (5)

 $\sigma _{i + 1}^2 = {\alpha _0} + {\alpha _1}{\left( {{\varepsilon _{2, i}} - \left( {\lambda + \gamma } \right)} \right)^2}\sigma _i^2 + \beta \sigma _i^2$ (6)

(4) 步骤4.运用方差互换产品定价模型和步骤3中模拟的路径推导出方差互换产品初始时刻价格为

 $V\left( {{S_0}, 0} \right) = M{{\rm{e}}^{ - rT}}\left[ {\sum\limits_{i = 1}^N {{{\left( {\ln \frac{{{S_{\left( 2 \right)i}}}}{{{S_{\left( 2 \right)i - 1}}}}} \right)}^2}} - K_{{\rm{Var}}}^2} \right]$

(5) 步骤5.计算V(b)=V-b(X-E(X))，b为一个可以估计出的常数，其中E(X)由下式给出：

 $\begin{array}{*{20}{c}} {W\left( {{S_0}, 0} \right) = {{\rm{e}}^{ - rT}}M\left[ {\sigma _{\rm{c}}^2T + \left( {{r^2} - r\sigma _{\rm{c}}^2 + } \right.} \right.}\\ {\left. {\left. {\frac{1}{4}\sigma _{\rm{c}}^4} \right)\sum\limits_{i = 1}^N {\Delta t_i^2 - K_{{\rm{Var}}}^2} } \right]} \end{array}$

(6) 步骤6.模拟m次(重复步骤1至5 m次)，并对每一次模拟所得的V(b)取平均得到GARCH模型下方差互换产品价格的控制变量估计为

 $\bar V\left( b \right) = \frac{1}{m}\sum\limits_{j = 1}^m {{V_j}\left( b \right)} = \bar V - b\left( {\bar X - E\left( X \right)} \right)$ (7)

 ${b^ * } \approx {{\hat b}_m} = \sum\limits_{j = 1}^m {\left( {{V_j} - \bar V} \right)\left( {{X_j} - \bar X} \right)} {\left[ {\sum\limits_{j = 1}^m {{{\left( {{X_j} - \bar X} \right)}^2}} } \right]^{ - 1}}$

 $R = \frac{1}{{\sqrt {1 - \rho _{XV}^2} }}$

 图 1 R与波动率的关系(N=1, m=5 000) Fig.1 Relation between the ratio and control variate parameter σc(N=1, m=5 000)

 ${\sigma _{\rm{c}}} = E\left( {\sigma \left( {{t_{i, j}}} \right)} \right) = \frac{1}{{m \times n}}\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^m {\sigma \left( {{t_{i, j}}} \right)} }$

X~N(μ, σ2)，则

 $\begin{array}{*{20}{c}} {E\left( {{{\rm{e}}^X}} \right) = {{\rm{e}}^{\mu + \frac{{{\sigma ^2}}}{2}}}}\\ {{\rm{Var}}\left( {{{\rm{e}}^X}} \right) = \left( {{{\rm{e}}^{{\sigma ^2}}} - 1} \right){{\rm{e}}^{2\mu + {\sigma ^2}}}} \end{array}$

 $\begin{array}{l} E\left( {{S_{\left( 1 \right)T}}} \right) = E\left( {{S_0}{{\rm{e}}^{\sum\limits_{i = 1}^n {\left( {r - \frac{1}{2}\sigma _{\rm{c}}^2} \right)\Delta t + {\sigma _{\rm{c}}}\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;\;{S_0}{{\rm{e}}^{rT}}{{\rm{e}}^{ - \frac{1}{2}\sigma _{\rm{c}}^2T}}E\left( {{{\rm{e}}^{\sum\limits_{i = 1}^n {{\sigma _{\rm{c}}}\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;\;{S_0}{{\rm{e}}^{rT}}{{\rm{e}}^{ - \frac{1}{2}\sigma _{\rm{c}}^2T}}{{\rm{e}}^{\frac{1}{2}\sigma _{\rm{c}}^2T}} = {S_0}{{\rm{e}}^{rT}} \end{array}$
 $\begin{array}{l} {\rm{Var}}\left( {{S_{\left( 1 \right)T}}} \right) = {\rm{Var}}\left( {{S_0}{{\rm{e}}^{\sum\limits_{i = 1}^n {\left( {r - \frac{1}{2}\sigma _{\rm{c}}^2} \right)\Delta t + {\sigma _{\rm{c}}}\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;S_0^2{{\rm{e}}^{2rT}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{i = 1}^n {{\sigma _{\rm{c}}}\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;S_0^2{{\rm{e}}^{2rT}}\left( {{{\rm{e}}^{\sigma _{\rm{c}}^2T}} - 1} \right) \end{array}$

 $\begin{array}{l} E\left( {{S_{\left( 2 \right)T}}} \right) = E\left( {{S_0}{{\rm{e}}^{\sum\limits_{i = 1}^n {\left( {r - \frac{1}{2}{\sigma ^2}\left( {{t_i}} \right)} \right)\Delta t + \sigma \left( {{t_i}} \right)\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;\;{S_0}{{\rm{e}}^{rT}}E\left( {{{\rm{e}}^{\sum\limits_{i = 1}^n {\left. { - \frac{1}{2}{\sigma ^2}\left( {{t_i}} \right)} \right)\Delta t + \sigma \left( {{t_i}} \right)\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) \end{array}$
 $\begin{array}{l} {\rm{Var}}\left( {{S_{\left( 2 \right)T}}} \right) = {\rm{Var}}\left( {{S_0}{{\rm{e}}^{\sum\limits_{i = 1}^n {\left( {r - \frac{1}{2}{\sigma ^2}\left( {{t_i}} \right)} \right)\Delta t + \sigma \left( {{t_i}} \right)\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2rT}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{i = 1}^n { - \frac{1}{2}{\sigma ^2}\left( {{t_i}} \right)\Delta t + {\sigma _{\rm{c}}}\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) \end{array}$

 ${\sigma ^ * } = E\left( {\sigma \left( {{t_{i, j}}} \right)} \right) = \frac{1}{{m \times n}}\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\sigma \left( {{t_{i, j}}} \right)} }$

 $\begin{array}{l} E\left( {{S_{\left( 2 \right)T}}} \right) = E\left( {{S_0}{{\rm{e}}^{\sum\limits_{i = 1}^n {\left( {r - \frac{1}{2}{{\left( {{\sigma ^ * }} \right)}^2}} \right)\Delta t + {\sigma ^ * }\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;\;{S_0}{{\rm{e}}^{rT}}{{\rm{e}}^{ - {{\left( {{\sigma ^ * }} \right)}^2}T}}E\left( {{{\rm{e}}^{\sum\limits_{i = 1}^n {{\sigma ^ * }\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;\;{S_0}{{\rm{e}}^{rT}}{{\rm{e}}^{ - {{\left( {{\sigma ^ * }} \right)}^2}T}}{{\rm{e}}^{{{\left( {{\sigma ^ * }} \right)}^2}T}} = \\ \;\;\;\;\;\;\;{S_0}{{\rm{e}}^{rT}} = E\left( {S_T^{\left( 1 \right)}} \right) \end{array}$
 $\begin{array}{l} {\rm{Var}}\left( {{S_{\left( 2 \right)T}}} \right) = {\rm{Var}}\left( {{S_0}{{\rm{e}}^{\sum\limits_{i = 1}^n {\left( {r - \frac{1}{2}{{\left( {{\sigma ^ * }} \right)}^2}} \right)\Delta t + {\sigma ^ * }\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2rT}}{{\rm{e}}^{ - {{\left( {{\sigma ^ * }} \right)}^2}T}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{i = 1}^n {{\sigma ^ * }\sqrt {\Delta t} {\varepsilon _{1, i}}} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2rT}}\left( {{{\rm{e}}^{{{\left( {{\sigma ^ * }} \right)}^2}T}} - 1} \right) \end{array}$

 $\left\{ \begin{array}{l} E\left( {{S_{\left( 1 \right)T}}} \right) = E\left( {{S_{\left( 2 \right)T}}} \right)\\ {\rm{Var}}\left( {{S_{\left( 1 \right)T}}} \right) = {\rm{Var}}\left( {{S_{\left( 2 \right)T}}} \right) \end{array} \right.$

3 数值计算模拟结果

 $\mathop {\max }\limits_{{\sigma _{\rm{c}}}} \left\{ {\frac{{{\rm{Var}}\left[ {{V_j}} \right]}}{{\mathop {\min }\limits_b \left\{ {{\rm{Var}}\left[ {{V_j}\left( b \right)} \right]} \right\}}}} \right\}$

 图 2 R与波动率的关系(N=1，m=2 000) Fig.2 Relation between R and volatility (N=1, m=2 000)

 图 3 R与波动率关系(N=10，m=2 000) Fig.3 Relation between R and volatility (N=10, m=2 000)
4 进一步的讨论

 $\frac{{{\rm{d}}{S_{\rm{t}}}}}{{{S_{\rm{t}}}}} = r{\rm{d}}t + {\sigma _{i{\rm{c}}}}{\rm{d}}{B_{\rm{t}}}$ (8)

 $\left\{ \begin{array}{l} {S_{\left( 1 \right)}}\left( {{t_0}} \right) = {S_0}\\ {S_{\left( 1 \right)}}\left( {{t_{i + 1}}} \right) = {S_{\left( 1 \right)}}\left( {{t_i}} \right){{\rm{e}}^{\sum\limits_{k = i + 1}^{p\left( {i + 1} \right)} {\left( {r - \frac{1}{2}\sigma _{i + 1, {\rm{c}}}^2} \right)\Delta t} }} \cdot \\ \;\;\;\;\;\;{{\rm{e}}^{\sum\limits_{k = i + 1}^{p\left( {i + 1} \right)} {{\sigma _{i + 1, {\rm{c}}}}\sqrt {\Delta t} {\varepsilon _{1, i + 1, k}}} }} \end{array} \right.$ (9)

GARCH模型的价格变化过程为

 $\left\{ \begin{array}{l} {S_{\left( 2 \right)}}\left( {{t_0}} \right) = {S_0}\\ {S_{\left( 2 \right)}}\left( {{t_{i + 1}}} \right) = {S_{\left( 2 \right)}}\left( {{t_i}} \right){{\rm{e}}^{\sum\limits_{k = i + 1}^{p\left( {i + 1} \right)} {\left( {r - \frac{1}{2}{\sigma ^2}\left( {{t_i} + k\Delta t} \right)} \right)\Delta t} }} \cdot \\ \;\;\;\;\;\;{{\rm{e}}^{\sum\limits_{k = i + 1}^{p\left( {i + 1} \right)} {\sigma \left( {{t_i} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, i + 1, k}}} }} \end{array} \right.$ (10)

 $\sigma _{i, {\rm{c}}}^2 = \frac{{{{\left( {E\left( {\sigma \left( {{t_{j, i}}} \right)} \right)} \right)}^2}{t_i} - {{\left( {E\left( {\sigma \left( {{t_{j, i - 1}}} \right)} \right)} \right)}^2}{t_{i - 1}}}}{{{t_i} - {t_{i - 1}}}}$

 $E\left( {\sigma \left( {{t_{j, i}}} \right)} \right) = \frac{1}{{m \times ip}}\sum\limits_{j = 1}^m {\sum\limits_{k = 1}^{ip} {\sigma \left( {{t_0} + k\Delta t} \right)} }$

 $\begin{array}{*{20}{c}} {E\left( {{{\rm{e}}^X}} \right) = {{\rm{e}}^{\mu + \frac{{{\sigma ^2}}}{2}}}}\\ {{\rm{Var}}\left( {{{\rm{e}}^X}} \right) = \left( {{{\rm{e}}^{{\sigma ^2}}} - 1} \right){{\rm{e}}^{2\mu + {\sigma ^2}}}} \end{array}$

 $\begin{array}{l} E\left( {{S_{\left( 1 \right)}}\left( {{t_1}} \right)} \right) = E\left( {{S_0}{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( {r - \frac{1}{2}{{\left( {\sigma _{1, {\rm{c}}}^2} \right)}^2}} \right)\Delta t} + \sum\limits_{k = 1}^p {\left( {\sigma _{1, {\rm{c}}}^2\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ {S_0}{{\rm{e}}^{r{t_1}}}E\left( {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( { - \frac{1}{2}\sigma _{1, {\rm{c}}}^2} \right)\Delta t} + \sum\limits_{k = 1}^p {\left( {\sigma _{1, {\rm{c}}}^2\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ {S_0}{{\rm{e}}^{r{t_2}}}E\left( {{{\rm{e}}^{ - \frac{1}{2}\sigma _{1, {\rm{c}}}^2{t_1} + {\sigma _{1, {\rm{c}}}}\sqrt {\Delta t} \sum\limits_{k = 1}^p {\left( {{\varepsilon _{1, 1, k}}} \right)} }}} \right) \end{array}$

 $\begin{array}{l} E\left( {{S_{\left( 1 \right)}}\left( {{t_1}} \right)} \right) = {S_0}{{\rm{e}}^{r{t_1}}}E\left( {{{\rm{e}}^{ - \frac{1}{2}\sigma _{1, {\rm{c}}}^2 + {\sigma _{1, {\rm{c}}}}\sqrt {\Delta t} \sum\limits_{k = 1}^p {\left( {{\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;{S_0}{{\rm{e}}^{r{t_1}}}{{\rm{e}}^{ - \frac{1}{2}\sigma _{1, {\rm{c}}}^2{t_1}}}{{\rm{e}}^{\frac{1}{2}\sigma _{1, {\rm{c}}}^2{t_1}}} = {S_0}{{\rm{e}}^{r{t_1}}} \end{array}$
 $\begin{array}{l} {\rm{Var}}\left( {{S_{\left( 1 \right)}}\left( {{t_1}} \right)} \right) = \\ \;\;\;\;\;\;\;{\rm{Var}}\left( {{S_0}{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( {r - \frac{1}{2}{{\left( {\sigma _{1, {\rm{c}}}^2} \right)}^2}} \right)\Delta t} + \sum\limits_{k = 2}^p {\left( {\sigma _{1, {\rm{c}}}^2\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_1}}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( { - \frac{1}{2}\sigma _{1, {\rm{c}}}^2} \right)\Delta t} + \sum\limits_{k = 1}^p {\left( {\sigma _{1, {\rm{c}}}^2\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_1}}}{{\rm{e}}^{ - \sigma _{1, {\rm{c}}}^2{t_1}}}{\rm{Var}}\left( {{{\rm{e}}^{{\sigma _{1, {\rm{c}}}}\sqrt {\Delta t} \sum\limits_{k = 1}^p {\left( {{\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_1}}}{{\rm{e}}^{ - \sigma _{1, {\rm{c}}}^2{t_1}}}{{\rm{e}}^{\sigma _{1, {\rm{c}}}^2{t_1}}}\left( {{{\rm{e}}^{\sigma _{1, {\rm{c}}}^2{t_1}}} - 1} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_1}}}\left( {{{\rm{e}}^{\sigma _{1, {\rm{c}}}^2{t_1}}} - 1} \right) \end{array}$

 $\begin{array}{l} E\left( {{S_{\left( 2 \right)}}\left( {{t_1}} \right)} \right) = E\left( {{S_0}{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( {r - \frac{1}{2}{\sigma ^2}\left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;{S_0}E\left( {{{\rm{e}}^{\sum\limits_{k = 1}^p {r\Delta t} + \sum\limits_{k = 1}^p {\left( { - \frac{1}{2}{\sigma ^2}\left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;{S_0}{{\rm{e}}^{r{t_1}}}E\left( {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( { - \frac{1}{2}\sigma \left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\left. {{e^{\sum\limits_{k = 1}^p {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ {\rm{Var}}\left( {{S_{\left( 2 \right)}}\left( {{t_1}} \right)} \right) = S_0^2{{\rm{e}}^{2r{t_1}}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( { - \frac{1}{2}{\sigma ^2}\left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) \end{array}$

 $E\left( {\sigma \left( {{t_{0, 1}}} \right)} \right) = \frac{1}{p}\sum\limits_{k = 1}^p {\sigma \left( {{t_0}\left| {k\Delta t} \right.} \right)}$

 $\begin{array}{l} E\left( {{S_{\left( 2 \right)}}\left( {{t_1}} \right)} \right) = {S_0}{{\rm{e}}^{r{t_1}}}E\left( {{{\rm{e}}^{ - \frac{1}{2}{{\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)} \right)}^2}{t_1}}} \cdot } \right.\\ \;\;\;\;\;\;\;\;\left. {{{\rm{e}}^{E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)\sqrt {\Delta t} \sum\limits_{k = 1}^p {{\varepsilon _{1, 1, k}}} }}} \right) = \\ \;\;\;\;\;\;\;\;{S_0}{{\rm{e}}^{r{t_1}}}{{\rm{e}}^{ - \frac{1}{2}{{\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)} \right)}^2}{t_1}}}{{\rm{e}}^{\frac{1}{2}{{\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)} \right)}^2}{t_1}}} = \\ \;\;\;\;\;\;\;\;{S_0}{{\rm{e}}^{r{t_1}}} = E\left( {{S_{\left( 1 \right)}}\left( {{t_1}} \right)} \right) \end{array}$
 $\begin{array}{l} {\rm{Var}}\left( {{S_{\left( 2 \right)}}\left( {{t_1}} \right)} \right) = S_0^2{{\rm{e}}^{2r{t_1}}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( { - \frac{1}{2}{\sigma ^2}\left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_1}}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( { - \frac{1}{2}{{\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)} \right)}^2}} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^p {\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} \right.} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_1}}}{{\rm{e}}^{ - {{\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)} \right)}^2}{t_1}}} \cdot \\ \;\;\;\;\;\;\;{\rm{Var}}\left( {{{\rm{e}}^{E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)\sqrt {\Delta t} \sum\limits_{k = 1}^p {{\varepsilon _{1, 1, k}}} }}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_1}}}{{\rm{e}}^{ - {{\left( {E\left( {\sigma \left( {{t_{0, 1}}} \right)} \right)} \right)}^2}{t_1}}} \cdot \\ \;\;\;\;\;\;\;{{\rm{e}}^{ - {{\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)} \right)}^2}{t_1}}}\left( {{{\rm{e}}^{\left. {{{\left( {E\left( {\sigma \left( {{t_{0, 1}}} \right)} \right)} \right)}^2}{t_1} - 1} \right)}}} \right) = \\ \;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_1}}}\left( {{{\rm{e}}^{{{\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)} \right)}^2}{t_1} - 1}}} \right) \end{array}$

 $\left\{ \begin{array}{l} E\left( {{S_{\left( 2 \right)}}\left( {{t_1}} \right)} \right) = E\left( {{S_{\left( 1 \right)}}\left( {{t_1}} \right)} \right)\\ {\rm{Var}}\left( {{S_{\left( 2 \right)}}\left( {{t_1}} \right)} \right) = {\rm{Var}}\left( {{S_{\left( 1 \right)}}\left( {{t_1}} \right)} \right) \end{array} \right.$

 $\left\{ \begin{array}{l} E\left( {{S_{\left( 1 \right)}}\left( {{t_2}} \right)} \right) = {S_0}{{\rm{e}}^{r{t_2}}}\\ {\rm{Var}}\left( {{S_{\left( 1 \right)}}\left( {{t_2}} \right)} \right) = S_0^2{{\rm{e}}^{2r{t_2}}}\left( {{{\rm{e}}^{ - \sigma _{1, {\rm{c}}}^2{t_1} - \sigma _{2, {\rm{c}}}^2\left( {{t_2} - {t_1}} \right)}} - 1} \right) \end{array} \right.$

 $\left\{ \begin{array}{l} E\left( {{S_{\left( 2 \right)}}\left( {{t_2}} \right)} \right) = {S_0}{{\rm{e}}^{r{t_2}}}E\left( {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( { - \frac{1}{2}{\sigma ^2}\left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right)\\ {\rm{Var}}\left( {{S_{\left( 2 \right)}}\left( {{t_2}} \right)} \right) = S_0^2{{\rm{e}}^{2r{t_2}}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( { - \frac{1}{2}{\sigma ^2}\left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) \end{array} \right.$

 $E\left( {\sigma \left( {{t_{j, 2}}} \right)} \right. = \frac{1}{{m \times 2p}}\sum\limits_{j = 1}^m {\sum\limits_{k = 1}^{2p} {\sigma \left( {{t_0} + k\Delta t} \right)} }$

 $\begin{array}{l} E\left( {{S_{\left( 2 \right)}}\left( {{t_2}} \right)} \right) = {S_0}{{\rm{e}}^{r{t_2}}}E\left( {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( { - \frac{1}{2}{\sigma ^2}\left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;\;{S_0}{{\rm{e}}^{r{t_2}}}E\left( {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( { - \frac{1}{2}{{\left( {E\left( {\sigma \left( {{t_{j, 2}}} \right)} \right)} \right)}^2}} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( {E\left( {\sigma \left( {{t_{j, 2}}} \right)} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;\;{S_0}{{\rm{e}}^{r{t_2}}}{{\rm{e}}^{ - \frac{1}{2}{{\left( {E\left( {\sigma \left( {{t_{j, 2}}} \right)} \right)} \right)}^2}{t_2}}} \cdot {{\rm{e}}^{\frac{1}{2}{{\left( {E\left( {\sigma \left( {{t_{j, 2}}} \right)} \right)} \right)}^2}{t_2}}} = {S_0}{{\rm{e}}^{r{t_2}}} \end{array}$
 $\begin{array}{l} {\rm{Var}}\left( {{S_{\left( 2 \right)}}\left( {{t_2}} \right)} \right) = S_0^2{{\rm{e}}^{2r{t_2}}}{\rm{Var}}\left( {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( { - \frac{1}{2}{\sigma ^2}\left( {{t_0} + k\Delta t} \right)} \right)\Delta t} }} \cdot } \right.\\ \;\;\;\;\;\;\;\;\left. {{{\rm{e}}^{\sum\limits_{k = 1}^{2p} {\left( {\sigma \left( {{t_0} + k\Delta t} \right)\sqrt {\Delta t} {\varepsilon _{1, 1, k}}} \right)} }}} \right) = \\ \;\;\;\;\;\;\;\;S_0^2{{\rm{e}}^{2r{t_2}}}\left( {{{\rm{e}}^{ - {{\left( {E\left( {\sigma \left( {{t_{j, 2}}} \right)} \right)} \right)}^2}{t_2}}} - 1} \right) \end{array}$

 $\left\{ \begin{array}{l} E\left( {{S_{\left( 2 \right)}}\left( {{t_2}} \right)} \right) = E\left( {{S_{\left( 1 \right)}}\left( {{t_2}} \right)} \right)\\ {\rm{Var}}\left( {{S_{\left( 2 \right)}}\left( {{t_2}} \right)} \right) = {\rm{Var}}\left( {{S_{\left( 1 \right)}}\left( {{t_2}} \right)} \right) \end{array} \right.$

 $- \sigma _{1, {\rm{c}}}^2{t_1} - \sigma _{2, {\rm{c}}}^2\left( {{t_2} - {t_1}} \right) = - {\left( {E\left( {\sigma \left( {{t_{0, 2}}} \right)} \right)} \right)^2}{t_2}$

 $\begin{array}{*{20}{c}} {\sigma _{2, {\rm{c}}}^2\left( {{t_2} - {t_1}} \right) = {{\left( {E\left( {\sigma \left( {{t_{j, 2}}} \right)} \right)} \right)}^2}{t_2} - \sigma _{1, {\rm{c}}}^2{t_1} = }\\ {{{\left( {E\left( {\sigma \left( {{t_{j, 2}}} \right)} \right)} \right)}^2}{t_2} - {{\left( {E\left( {\sigma \left( {{t_{j, 1}}} \right)} \right)} \right)}^2}{t_1}} \end{array}$

 $\sigma _{i, {\rm{c}}}^2\left( {{t_i} - {t_{i - 1}}} \right) = {\left( {E\left( {\sigma \left( {{t_{j, i}}} \right)} \right)} \right)^2}{t_i} - {\left( {E\left( {\sigma \left( {{t_{j, i - 1}}} \right)} \right)} \right)^2}{t_{i - 1}}$

5 结论

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