几种指数的波动率分析

论文总字数：20272字

摘 要

随着金融市场的不断发展和更多金融产品的出现，金融市场的各种模型也相继被提出。其中有一个非常重要的就是BS定价模型，自从BS定价模型被提出来之后，人们将假定为常数的波动率反带入了BS定价模型却发现了波动率微笑的现象，从此人们认为指数波动率为常数的这一想法被推翻了，波动率的预测就成为了金融市场的一个重大问题。金融市场的波动可以通过各种金融指数的波动率很好的去体现，波动率的预测帮助投资者合理预见市场风险，也影响着期权定价的准确性，同时还影响着期权的收益问题。更严重的，如果对市场的风险未知的情况下进行了一些不合理的操作，甚至产生很大的金融危机。所以本文想借助随机过程和随机波动模型参数预测所运用到的MCMC方法对金融指数的波动率进行分析，合理进行预测。

本文首先介绍了股票指数和波动率，初步了解波动率之后，将会具体介绍波动率分类中的历史波动率和隐含波动率，其中穿插了要用到的我们重要的BS公式的内容。接下来介绍的是用于连续时间模型参数估计的MCMC，以及MCMC的两种基本采样方式：吉布斯抽样和MH抽样。最后会设计到实际的数据计算，首先是关于常见的三种指数：上证指数，道琼斯指数，标普500指数的历史波动率的计算及分析；其次是关于上证50ETF指数的隐含波动率的计算和分析；最后是用MCMC方法联系了指数波动率并且对美国芝加哥商品期权交易所提供的数据进行了处理和分析。

关键词：波动率，指数，MCMC方法

ABSTRACT

With the continuous development and progress of financial market and the emergence of mo

re financial products, various models of financial market have been put forward. One of them is the BS pricing model. Since the BS pricing model was put forward, people have brought the assumed constant volatility back into the BS pricing model but found the phenomenon of volatility smiling. From then on, the idea that the index volatility is constant has been overturned, and the prediction of volatility has become a major problem in the financial market. The volatility of financial market can be well reflected by the volatility of various financial indices. The prediction of volatility helps investors to predict market risks reasonably, and also affects the accuracy of option pricing, as well as the return of options. More seriously, if the market risk is unknown, some unreasonable operations will be carried out, and even a great financial crisis will occur. Therefore, this paper intends to use the MCMC method used in the parameter prediction of stochastic process and stochastic volatility model to analyze the volatility of financial index and make a reasonable prediction.

Firstly, this paper introduces stock index and volatility. After a preliminary understanding of volatility, we will introduce the historical volatility and implied volatility in volatility classification, which are interspersed with the content of our important BS formula. Next, MCMC for parameter estimation of continuous time model is introduced, and two basic sampling methods of MCMC are introduced: Gibbs sampling and MH sampling. Finally, the actual data calculation will be designed. First, the calculation and analysis of the historical volatility of the Shanghai Stock Exchange Index, the Dow Jones Index and the Samp;P 500 Index. Second, the calculation and analysis of the implied volatility of the Shanghai 50ETF Index. Finally, the MCMC method is used to link the index volatility and to improve the data provided by the Chicago Commodity Options Exchange. Processing and analysis were carried out.

Key words: Volatility,index,MCMC method

目 录

一、 股票指数 4

二、 波动率 5

三、 历史波动率 6

四、 布莱克-斯科尔斯-默顿定价公式（BS公式） 7

五、 隐含波动率 9

六、 MCMC方法和波动率的平稳分布 10

6.1 Gibbs抽样 11

6.2 MH抽样 11

结论 14

7.1 历史波动率的计算 14

7.2 隐含波动率的计算 16

7.3 关于MCMC和隐含波动分布的应用 19

谢 辞 24

参考文献 25

附 录 26

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