﻿ 基于混合分数布朗运动下带跳的两值期权定价

# 基于混合分数布朗运动下带跳的两值期权定价Binary Option Pricing with Jump Based on Mixed Fractional Brownian Motion

Abstract: In this paper, we study the pricing problem of binary options in which stock price obeys the mixed fraction Brownian motion model with jump. First, the pricing formula of binary options is obtained through the heat conduction equation theory, and then the series solution formula of binary options is obtained by the insurance actuarial method. Then, the numerical solution is obtained by means of Monte Carlo simulation and finite difference method. Using the pricing formula obtained by the heat conduction equation as the standard, the options price obtained by the other three methods is compared with the pricing formula solution, and the results are compared. The feasibility and validity of this method are obtained.

1. 引言

2. 预备知识

2.1. 基本符号及引理

S为股票现价，X为期权的执行价格， $\mu$ 为预期收益率， $\sigma$ 为无跳跃时股价波动率，r为无风险率，T为期权的到期日t为当前时刻V为t时刻基于价格为S的股票的衍生证券价格。

2.2. 两值期权

$H\left({S}_{T},T\right)=\left\{\begin{array}{l}Q,\text{}{S}_{T}>0.\\ 0,\text{}{S}_{T}\le 0.\end{array}$

$N\left({S}_{T},T\right)=\left\{\begin{array}{l}{S}_{T},\text{}{S}_{T}>0.\\ 0,\text{}{S}_{T}\le 0.\end{array}$

3. 数学模型及定价公式

$\text{d}S\left(t\right)=\left(\mu \left(t\right)-\lambda k\right)S\left(t\right)\text{d}t+{\sigma }_{H}S\left(t\right)\text{d}{B}_{t}^{H}+\sigma S\left(t\right)\text{d}{B}_{t}+US\left(t\right)\text{d}{N}_{t}.$ (1)

3.1. 微分方程法

$\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}\left[{\sigma }^{2}+2{\sigma }_{H}^{2}H{t}^{2H-1}+\lambda {k}^{2}\right]{S}^{2}\frac{{\partial }^{2}V}{\partial {S}^{2}}-rV=0$

$\sigma {1}^{2}={\sigma }^{2}+\lambda {k}^{2}$

$\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}\left[\sigma {1}^{2}+2{\sigma }_{H}^{2}H{t}^{2H-1}\right]{S}^{2}\frac{{\partial }^{2}V}{\partial {S}^{2}}-rV=0$ (2)

$V={S}_{t}N\left(d1\right)$

$d1=\frac{\mathrm{ln}\frac{{S}_{t}}{X}+r\left(T-t\right)+\frac{1}{2}\sigma {1}^{2}\left(T-t\right)+\frac{1}{2}{\sigma }_{H}^{2}\left({T}^{2H}-{t}^{2H}\right)}{\sqrt{\sigma {1}^{2}\left(T-t\right)+{\sigma }_{H}^{2}\left({T}^{2H}-{t}^{2H}\right)}}$

$V=Q{\text{e}}^{-r\left(T-t\right)}N\left(d2\right)$

$d2=d1-\sqrt{\sigma {1}^{2}\left(T-t\right)+{\sigma }_{H}^{2}\left({T}^{2H}-{t}^{2H}\right)}$

3.2. 保险精算法

$S\left(t\right)=S\left(0\right)\mathrm{exp}\left\{{\int }_{0}^{t}\left[\mu \left(s\right)-\lambda k-H{\sigma }_{H}^{2}{s}^{2H-1}-\frac{1}{2}{\sigma }^{2}\right]\text{d}s+{\int }_{0}^{t}{\sigma }_{H}\text{d}{B}_{t}^{H}+{\int }_{0}^{t}\sigma \text{d}{B}_{t}+\underset{i=0}{\overset{{N}_{t}}{\sum }}\mathrm{ln}\left(1+{U}_{i}\right)\right\}.$

$\mathrm{exp}\left\{{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right\}=\frac{E\left[S\left(t\right)\right]}{S\left(0\right)}.$

$\mathrm{exp}\left\{-{\int }_{0}^{T}\beta \left(s\right)\text{d}s\right\}S\left(T\right)>\mathrm{exp}\left\{-{\int }_{0}^{T}r\left(s\right)\text{d}s\right\}X.$

$\mathrm{exp}\left\{-{\int }_{0}^{T}\beta \left(s\right)\text{d}s\right\}S\left(T\right)<\mathrm{exp}\left\{-{\int }_{0}^{T}r\left(s\right)\text{d}s\right\}X.$

$E\left[{\text{e}}^{Y}{I}_{\left\{Y>a\right\}}\right]={\text{e}}^{{\mu }_{Y}+\frac{{\sigma }_{Y}^{2}}{2}}N\left(\frac{-a+{\sigma }_{Y}^{2}+{\mu }_{Y}}{{\sigma }_{Y}}\right).$

$\begin{array}{l}c1=S\left(0\right)\underset{n=0}{\overset{+\infty }{\sum }}\frac{{\left(\lambda \left(1+k\right)T\right)}^{n}{\text{e}}^{-\lambda \left(1+k\right)T}}{n!}N\left(d1\right)\\ d1=\frac{-\mathrm{ln}\frac{X}{S\left(0\right)}+{\int }_{0}^{T}r\left(s\right)\text{d}s+{\mu }_{n}+{\sigma }_{n}}{\sqrt{{\sigma }_{n}}}\\ d2=\frac{-\mathrm{ln}\frac{X}{S\left(0\right)}+{\int }_{0}^{T}r\left(s\right)\text{d}s+{\mu }_{n}}{\sqrt{{\sigma }_{n}}}=d1-\sqrt{{\sigma }_{n}}\end{array}$

${\int }_{0}^{t}\beta \left(s\right)\text{d}s={\int }_{0}^{t}r\left(s\right)\text{d}s.$

$\begin{array}{c}{\mathrm{e}}^{-{\int }_{0}^{T}\beta \left(s\right)\text{d}s}S\left(T\right)=S\left(0\right){\text{e}}^{\left\{{\int }_{0}^{T}\left(\mu \left(s\right)-\lambda k-H{\sigma }_{H}^{2}{s}^{2H-1}-\frac{1}{2}{\sigma }^{2}\right)\text{d}s+{\int }_{0}^{T}{\sigma }_{H}\text{d}{B}_{t}^{H}+{\int }_{0}^{T}\sigma \text{d}{B}_{t}+\underset{i=0}{\overset{{N}_{T}}{\sum }}\mathrm{ln}\left(1+{U}_{i}\right)\right\}}\\ =S\left(0\right){\text{e}}^{\left\{{\int }_{0}^{T}\left(-\lambda k-H{\sigma }_{H}^{2}{s}^{2H-1}-\frac{1}{2}{\sigma }^{2}\right)\text{d}s+{\int }_{0}^{T}{\sigma }_{H}\text{d}{B}_{t}^{H}+{\int }_{0}^{T}\sigma \text{d}{B}_{t}+\underset{i=0}{\overset{{N}_{T}}{\sum }}\mathrm{ln}\left(1+{U}_{i}\right)\right\}}\\ =S\left(0\right){\text{e}}^{{X}_{{N}_{T}}}\end{array}$ (3)

${X}_{n}={\int }_{0}^{T}\left(-\lambda k-H{\sigma }_{H}^{2}{s}^{2H-1}-\frac{1}{2}{\sigma }^{2}\right)\text{d}s+{\int }_{0}^{T}{\sigma }_{H}\text{d}{B}_{t}^{H}+{\int }_{0}^{T}\sigma \text{d}{B}_{t}+\underset{i=0}{\overset{n}{\sum }}\mathrm{ln}\left(1+{U}_{i}\right).$

$\begin{array}{l}{\int }_{0}^{T}{\sigma }_{H}\text{d}{B}_{t}^{H}\sim N\left(0,{\int }_{0}^{T}2H{\sigma }_{H}^{2}{s}^{2H-1}\text{d}s\right),\\ {\int }_{0}^{T}\sigma \text{d}{B}_{t}\sim N\left(0,{\int }_{0}^{T}{\sigma }^{2}\text{d}s\right),\\ \underset{i=0}{\overset{n}{\sum }}\mathrm{ln}\left(1+{U}_{i}\right)\sim N\left(n\mathrm{ln}\left(1+k\right)-\frac{n{\sigma }_{J}^{2}}{2},n{\sigma }_{J}^{2}\right)\end{array}$

${\mu }_{n}={\int }_{0}^{T}\left(-\lambda k-H{\sigma }_{H}^{2}{s}^{2H-1}-\frac{1}{2}{\sigma }^{2}\right)\text{d}s+n\mathrm{ln}\left(1+k\right)-\frac{n{\sigma }_{J}^{2}}{2},\text{\hspace{0.17em}}{\sigma }_{n}={\int }_{0}^{T}2H{\sigma }_{H}^{2}{s}^{2H-1}\text{d}s+{\int }_{0}^{T}{\sigma }^{2}\text{d}s+n{\sigma }_{J}^{2}$

${\mathrm{e}}^{-{\int }_{0}^{T}\beta \left(s\right)\text{d}s}S\left(T\right)>{\mathrm{e}}^{-{\int }_{0}^{T}r\left(s\right)\text{d}s}X⇔{X}_{N\left(T\right)}>\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s.$

$\begin{array}{c}c1=E\left[{\mathrm{e}}^{-{\int }_{0}^{T}\beta \left(s\right)\text{d}s}S\left(T\right){I}_{\left\{{\mathrm{e}}^{-{\int }_{0}^{T}\beta \left(s\right)\text{d}s}S\left(T\right)>{\mathrm{e}}^{-{\int }_{0}^{T}r\left(s\right)\text{d}s}X\right\}}\right]\\ =E\left[{\mathrm{e}}^{-{\int }_{0}^{T}\beta \left(s\right)\text{d}s}S\left(T\right){I}_{\left\{{X}_{N\left(T\right)}>\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s\right\}}\right]\\ =E\left[S\left(0\right){\mathrm{e}}^{{X}_{n}}{I}_{\left\{{X}_{n}>\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s\right\}}|N\left(T\right)=n\right]\\ =S\left(0\right)\underset{n=0}{\overset{+\infty }{\sum }}\frac{{\left(\lambda T\right)}^{n}{\text{e}}^{-\lambda T}}{n!}E\left[{\mathrm{e}}^{{X}_{n}}{I}_{\left\{{X}_{n}>\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s\right\}}\right]\end{array}$

$\begin{array}{c}E\left[{\mathrm{e}}^{{X}_{n}}{I}_{\left\{{X}_{n}>\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s\right\}}\right]={\text{e}}^{{\mu }_{n}+\frac{{\sigma }_{n}}{2}}N\left(\frac{-\left(\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s\right)+{\sigma }_{n}^{2}+{\mu }_{n}}{{\sigma }_{n}}\right)\\ ={\text{e}}^{{\mu }_{n}+\frac{{\sigma }_{n}}{2}}N\left(\frac{-\left(\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s\right)+{\sigma }_{n}+{\mu }_{n}}{\sqrt{{\sigma }_{n}}}\right)\\ ={\left(1+k\right)}^{n}{\text{e}}^{-\lambda kT}N\left(d1\right)\end{array}$

$\begin{array}{c}c1=E\left[S\left(0\right){\mathrm{e}}^{{X}_{N\left(T\right)}}{I}_{\left\{{X}_{N\left(T\right)}>\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s\right\}}\right]\\ =S\left(0\right)\underset{n=0}{\overset{+\infty }{\sum }}\frac{{\left(\lambda T\right)}^{n}{\text{e}}^{-\lambda T}}{n!}{\left(1+k\right)}^{n}{\text{e}}^{-\lambda kT}N\left(d1\right)\\ =S\left(0\right)\underset{n=0}{\overset{+\infty }{\sum }}\frac{{\left(\lambda \left(1+k\right)T\right)}^{n}{\text{e}}^{-\lambda \left(1+k\right)T}}{n!}N\left(d1\right)\end{array}$

$\begin{array}{c}c1=E\left[{\mathrm{e}}^{-{\int }_{0}^{T}\beta \left(s\right)\text{d}s}S\left(T\right){I}_{\left\{{X}_{N\left(T\right)}>\mathrm{ln}\frac{X}{S\left(0\right)}-{\int }_{0}^{T}r\left(s\right)\text{d}s\right\}}\right]\\ =S\left(0\right)\underset{n=0}{\overset{+\infty }{\sum }}\frac{{\left(\lambda \left(1+k\right)T\right)}^{n}{\text{e}}^{-\lambda \left(1+k\right)T}}{n!}N\left(d1\right)\end{array}$

$c2=Q\underset{n=0}{\overset{+\infty }{\sum }}\frac{{\left(\lambda T\right)}^{n}{\text{e}}^{-\lambda T}}{n!}N\left(d2\right).$

4. 数值解法

4.1. 有限差分法

4.1.1. 资产或无值看涨期权

$\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}\left[{\sigma }^{2}+2{\sigma }_{H}^{2}H{t}^{2H-1}+\lambda {k}^{2}\right]{S}^{2}\frac{{\partial }^{2}V}{\partial {S}^{2}}-rV=0$ (4)

$\frac{\partial V}{\partial t}=\frac{{V}_{i,\text{}j+1}-{V}_{i,\text{}j}}{\Delta t}$ (5)

$\frac{\partial V}{\partial S}=\frac{{V}_{i+1,\text{}j}-{V}_{i-1,\text{}j}}{2\Delta S}$ (6)

$\frac{{\partial }^{2}V}{\partial {S}^{2}}=\frac{{V}_{i+1,\text{}j}+{V}_{i-1,\text{}j}-2{V}_{i,\text{}j}}{\Delta {S}^{2}}$ (7)

$\left\{\begin{array}{l}{a}_{i}{V}_{A}\left(i-1,j\right)+{b}_{i}{V}_{A}\left(i,j\right)+{c}_{i}{V}_{A}\left(i+1,j\right)={V}_{A}\left(i+1,j+1\right)\\ {V}_{A}\left(i,N\right)=N\left({S}_{T},T\right)\\ {V}_{A}\left(0,j\right)=0\\ {V}_{A}\left(M,j\right)={S}_{\mathrm{max}}\end{array}$ (8)

$\begin{array}{l}{a}_{i}=\frac{-\Delta t{L}_{i}+ri\Delta {S}^{2}\Delta t}{2\Delta {S}^{2}},\text{}{b}_{i}=\frac{\Delta t{L}_{i}+\Delta {S}^{2}\text{+}r\Delta {S}^{2}\Delta t}{\Delta {S}^{2}},\text{}\\ {\text{c}}_{i}=\frac{-\Delta t{L}_{i}-ri\Delta {S}^{2}\Delta t}{2\Delta {S}^{2}},\text{}{L}_{i}=\left({\sigma }^{2}+\lambda {k}^{2}+2{\sigma }_{H}^{2}H{t}^{2H-1}\right){\left(i\Delta S\right)}^{2}\end{array}$

${J}_{M-1}{T}_{M-1}={Z}_{M}$ (9)

${J}_{M-1}=\left[\begin{array}{ccccc}{b}_{1}& {c}_{1}& & & \\ {a}_{2}& {b}_{2}& {c}_{2}& & \\ & \ddots & \ddots & \ddots & \\ & & & & \\ & & & {a}_{M-1}& {b}_{M-1}\end{array}\right],\text{}{T}_{M-1}=\left[\begin{array}{c}{V}_{{A}_{1,N-1}}\\ {V}_{{A}_{2,N-1}}\\ ⋮\\ {V}_{{A}_{M-1,N-1}}\end{array}\right],\text{}{Z}_{M}=\left[\begin{array}{c}{V}_{{A}_{1,N}}-{a}_{1}{V}_{{A}_{0,N}}\\ {V}_{{A}_{2,N}}\\ ⋮\\ {V}_{{A}_{M-1,N}}-{c}_{M-1}{V}_{{A}_{M,N-1}}\end{array}\right]$

4.1.2. 现金或无值看涨期权

$\left\{\begin{array}{l}{a}_{i}{V}_{C}\left(i-1,j\right)+{b}_{i}{V}_{C}\left(i,j\right)+{c}_{i}{V}_{C}\left(i+1,j\right)={V}_{C}\left(i+1,j+1\right)\\ {V}_{C}\left(i,N\right)=H\left({S}_{T},T\right)\\ {V}_{C}\left(0,j\right)=0\\ {V}_{C}\left(M,j\right)=Q\end{array}$

4.2. 蒙特卡洛模拟

Boyle在1977年首次对欧式期权利用蒙特卡洛模拟的方法进行了定价，自此开始蒙特卡洛模拟方法在金融分析方面发挥了相当重要的作用，尤其在关于衍生产品定价和确定套期保值基本参数这些方面产生了重大作用，并且成为了一种有效的数值分析方法。

$0=t\left(0\right)

$i=0,1,2,\cdots ,N;\text{\hspace{0.17em}}t\left(i\right)=i\Delta t$。则得到如下的模拟路径：

${S}_{t\left(i\right)}={S}_{t\left(i-1\right)}{\text{e}}^{\left(\mu -\lambda k-H{\sigma }_{H}^{2}{t}^{2H-1}-\frac{1}{2}{\sigma }^{2}\right)\left(t\left(i\right)-t\left(i-1\right)\right)+\sigma \Delta {B}_{t}+{\sigma }_{H}\Delta {B}_{t}^{H}+\underset{i=0}{\overset{{N}_{t}\left(i\right)}{\sum }}\mathrm{ln}\left(1+{U}_{i}\right)}.$ (10)

1) 模拟布朗运动：产生服从标准正态分布的随机序列。

2) 模拟分数布朗运动：将满足标准正态分布的随机序列转化为分数布朗运动序列。

3) 我们模拟一个强度为 $\lambda \Delta t$ 的Poisson过程的随机数N。

4) 如果 $N=k$，我们模拟具有规则 $\mathrm{ln}\left(1+{U}_{i}\right)\sim N\left({\mu }_{J},{\sigma }_{J}^{2}\right)$$i=1,2,\cdots ,N$ 的N个独立随机变量。

5) 把上面模拟的随机序列带入式子(10)。

Table 1. Asset or valueless binary call option pricing under mixed fraction jump diffusion model

5. 数值算例

[1] 闫海峰, 刘三阳. 广义Black_Scholes模型期权定价新方法–保险精算方法[J]. 应用数学和力学, 2003, 24(7): 730-738.

[2] 李红. 跳跃–扩散模型下的期权定价[D]: [硕士学位论文]. 湖南: 湖南师范大学, 2006.

[3] 车韧, 何传江, 姚梅. 分形CEV模型及其蒙特卡罗模拟[J]. 重庆大学学报(自然科学版), 2007, 30(11): 148-151.

[4] 石方圆. 跳–扩散及O-U过程下期权的保险精算定价[D]: [硕士学位论文]. 河北: 河北师范大学, 2016.

[5] 吴云, 何建敏. 两值期权的定价模型及其求解研究[J]. 管理工程学报, 2002, 16(4): 108-110.

[6] 杜雪樵. CEV模型下两值期权的数值解[J]. 南方经济, 2006(2): 23-28.

[7] 杨珊, 薛红, 马惠馨. 分数跳–扩散下两值期权定价[J]. 四川理工学院学报(自然科学版), 2010, 23(4): 391-393.

[8] 袁国军. 基于半离散化的CEV过程下两值期权定价研究[J]. 系统工程学报, 2012, 27(1): 19-25.

[9] 王海叶. 两值期权定价公式的推广[J]. 三明学院学报, 2016, 33(4): 1-5.

[10] 丁华. B&P作用下两值期权的数值解[J]. 考试周刊, 2016(86): 53.

[11] 付培. 混合分数布朗运动下的两值期权定价模型[J]. 佛山科学技术学院学报(自然科学版), 2018, 36(2): 13-19.

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