# 基于二元期权组合的看涨宝收益凭证产品定价研究A Study on Pricing of Kanzhangbao Income Document Product Based on Binary Options

Abstract: In the international financial markets, the trading volume of options products grows at an in-creasing rate, and the position of options in the derivatives market will increase significantly. Both the designer and the investor of the option product should have a certain understanding of the pricing of options in order to better benefit from the options market. This paper first introduces the development of the financial derivatives market, which describes in detail the concept, classi-fication and advantages and disadvantages of the binary digital options and deduces the pricing method of binary option combination. After that, it briefly introduces the product of Kanzhangbao income document products and objectively detailed analyze the income structure of this product, the design of the indicators, and product features. Finally, the pricing function of this product is deduced by using the Black-Scholes option pricing model, and Monte-Carlo methods is used to si-mulate the pricing.

1. 绪论

1.1. 文献综述

1.2. 研究意义及文章结构

2016年年末，国内某券商为了增强自己在期权市场的竞争力，设计推出了看涨宝保本型收益凭证产品，本文将用蒙卡方法对此期权嵌入式收益凭证产品进行分析和定价研究，方便广大投资者对于此产品有更好的了解。文章安排内容如下：第一节简述已有的期权定价方法；第二节介绍二元期权的发展历史、相关概念以及优缺点，并详细推导了二元期权组合的定价过程；第三节介绍并分析此产品；第四节运用蒙卡方法对此产品进行定价；第五节总结全文并指出有待改进的不足之处。

2. 定价基础——二元期权组合定价

2.1. 概述

2.2. 二元期权定价

$\text{d}S=rS\left(t\right)\text{d}t+\sigma S\left(t\right)\text{d}W\left(t\right)$ (1)

$S\left(T\right)=S\left(0\right){\text{e}}^{\left(r-{\sigma }^{2}/2\right)T+\sigma W\left(t\right)}$ (2)

$S\left(T\right)=S\left(0\right){\text{e}}^{\left(r-{\sigma }^{2}/2\right)T-\sigma \sqrt{T}Z}$ (3)

$\text{payoff}=\left\{\begin{array}{l}C,\text{\hspace{0.17em}}\text{\hspace{0.17em}}S\left(T\right)>H\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$ (4)

$\text{price}=C{\text{e}}^{-rT}E\left[{I}_{S\left(T\right)>H}\left(S\left(T\right)\right)\right]$ (5)

$\text{price}=C{\text{e}}^{-rT}P\left\{S\left(T\right)>H\right\}$ (6)

$\text{price}=C{\text{e}}^{-rT}P\left\{Z<\frac{\mathrm{ln}\frac{S\left(0\right)}{H}+\left(r-{\sigma }^{2}/2\right)T}{\sigma \sqrt{T}}\right\}$ (7)

$\begin{array}{l}\text{price}=C{\text{e}}^{-rT}N\left(d\right)\\ d=\frac{\mathrm{ln}\frac{S\left(0\right)}{H}+\left(r-{\sigma }^{2}/2\right)T}{\sigma \sqrt{T}}\end{array}$ (8)

2.3. 二元期权组合定价

$\text{payoff}=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}S\left(T\right)<{H}_{1}\\ {C}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{H}_{1}\le S\left(T\right)<{H}_{2}\\ {C}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}S\left(T\right)\ge {H}_{2}\end{array}$ (9)

${\text{payoff}}_{\text{1}}=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}S\left(T\right)<{H}_{1}\\ {C}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$ (10)

${\text{payoff}}_{\text{2}}=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}S\left(T\right)<{H}_{2}\\ {C}_{2}-{C}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\end{array}$ (11)

Figure 1. Income structure of a combination of two cash or zero options

$\begin{array}{l}\text{price}={C}_{1}{\text{e}}^{-rT}N\left({d}_{1}\right)+\left({C}_{2}-{C}_{1}\right){\text{e}}^{-rT}N\left({d}_{2}\right)\\ {d}_{1}=\frac{\mathrm{ln}\frac{S\left(0\right)}{{H}_{1}}+\left(r-{\sigma }^{2}/2\right)T}{\sigma \sqrt{T}}\\ {d}_{2}=\frac{\mathrm{ln}\frac{S\left(0\right)}{{H}_{2}}+\left(r-{\sigma }^{2}/2\right)T}{\sigma \sqrt{T}}\end{array}$ (12)

3. 看涨宝收益凭证产品分析

3.1. 产品简

3.2. 产品分析

Figure 2. Income structure of Kanzhangbao income document products

Table 1. Income structure table of Kanzhangbao income document products

$V\left({P}_{T}\right)=\left\{\begin{array}{l}0%,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{T}<0%\\ 5%,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0%\le {P}_{T}<15%\\ 10%,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{T}\ge 15%\end{array}$ (13)

${V}_{1}\left({P}_{T}\right)=\left\{\begin{array}{l}0%,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{T}<0%\\ 5%,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{T}\ge 0%\end{array}$ (14)

${V}_{2}\left({P}_{T}\right)=\left\{\begin{array}{l}0%,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{T}<15%\\ 5%,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{P}_{T}\ge 15%\end{array}$ (15)

$V\left({P}_{T}\right)={V}_{1}\left({P}_{T}\right)+{V}_{2}\left({P}_{T}\right)$ (16)

Figure 3. Income structure of financial product of Formula (14)

Figure 4. Income structure of financial product of Formula (15)

4. 看涨宝收益凭证产品定价

4.1. 定价方法

$S\left(t\right)=S\left(0\right){\text{e}}^{\left(r-{\sigma }^{2}/2\right)t-\sigma \sqrt{t}Z}$ (17)

$V={\text{e}}^{-rT}\left\{\begin{array}{l}0%,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\in \left[0,T\right]使得S\left(t\right) (18)

4.2. 数据来源及说明

4.3. 产品定价编程思路

4.4. 蒙卡模拟结果及分析

2016年11月30日至2017年2月28日内中证500指数的具体收盘价如表2所示，2016年11月30日开盘价为6605.36，2016年12月1日开盘价为6593.59，2016年12月1日收盘价为6624.47，均高于

Figure 5. Closing price of the CSI 500 index

Table 2. Closing price of the CSI 500 index

5. 结论与展望

[1] Bachelier, L. (1900) Throrie de la speculation. Annales Normale Superieure, 17, 21-86.

[2] Sprenkle, C.M. (1964) Warrant Price as Idicators of Expections and Preferences in the Random Character of Stock Market Price. Cambridge MIT Press, London.

[3] Boness, J. (1964) Elements of a Theory of Stock Option Value. Journal of Political Economy, 72, 163-175.
https://doi.org/10.1086/258885

[4] Samuelson, P.A. (1965) Rational Theory of Theory of Warrant Pricing. Industrial Man-agement Review, 72, 163-175.

[5] Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654.
https://doi.org/10.1086/260062

[6] 周世军, 岳朝龙. 蒙特卡罗模拟在期权定价中的应用[J]. 安徽工业大学学报(社会科学版), 2009, 26(1): 24-25.

[7] 张丽虹. 欧式期权定价的蒙卡方法[J]. 经济研究导刊, 2015(15): 104-108 + 112.

[8] 王杨, 张寄洲, 傅毅. 双障碍期权的定价问题[J]. 上海师范大学学报(自然科学版), 2009, 38(4): 347-354.

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