随机波动率模型与VIX期权定价问题研究

论文总字数：31604字

摘 要

本文基于描述市场波动率的连续时间随机波动率模型，以看涨期权为例，运用随机过程、金融数学、金融市场等的相关知识，结合标的资产价格过程，来研究期权定价模型。

第一章，介绍了本文的研究背景，包括期权、VIX指数的含义及发展历程、经典的B-S模型和常见的连续时间随机波动率模型。

第二章，阐述了随机过程中的几个在期权定价问题中应用广泛的知识，依次是维纳过程、二次变差定理和伊藤公式。

第三章，根据经验研究表明，3/2模型被证明是捕获VIX指数行为能力最优的连续时间随机波动率模型，因此在本章节我们基于3/2模型、通过理论直接推导出解析解和寻找其满足的PDE这两种方式来寻找具有执行价格和到期时间的VIX看涨期权值。

第四章，为使结果更加具有现实性和普适性，在原有的3/2模型中引入泊松跳项，并进一步研究含泊松跳的VIX指数看涨期权的定价。

第五章，引入了标的资产价格过程，标的资产的价格也会影响看涨期权值，即此时。并且为了更符合真实情况，我们假设标的资产价格过程符合几何布朗运动，并同时引入了泊松跳项，然后进一步地以3/2模型和含泊松跳的标的资产价格过程为基础，利用对冲思想来寻找此时的看涨期权定价模型。

关键词：期权定价、VIX指数、随机波动率模型、泊松跳、标的资产价格过程

Abstract

Based on the continuous-time stochastic volatility models describing market volatility and the underlying asset price process, this paper uses the call option as an example to study the option pricing model by using the relevant knowledge of stochastic processes, financial mathematics and financial markets.

The first chapter firstly introduces the research background of this paper, such as the meaning and development of options and the VIX index. Secondly this chapter introduces the B-S model which is the important basis of option pricing problems. Then some common continuous-time stochastic volatility models are introduced.

The second chapter expounds the knowledge of some stochastic processes widely used in the option pricing problems—Levy process, Wiener process, quadratic variation theorem and Ito formula.

In the third chapter, empirical research shows that the 3/2 model is proved to be the continuous-time stochastic volatility model with the best ability to capture the VIX index. Therefore, in this chapter, we find the VIX call option value with the execution price X and the expiration time T by directly deriving the analytical solution through the theory and finding the PDE that it satisfies to.

In the fourth chapter, the Poisson jump is introduced into the original 3/2 model to make the results of this research more realistic and universal. And the pricing of the VIX index call option with Poisson jump is further studied.

In the fifth chapter, the underlying asset price process is introduced. The underlying asset’s price can also affect the price of call option, i.e. . At the same time, in order to fit the real situation better, the Poisson jump is introduced when the standard asset price process satisfies the Geometric Brown Motion(GBM), and the call option pricing model at this time is founded by using the hedging theory based on the 3/2 model and the underlying asset price process with Poisson jump.

KEY WORDS: option pricing, VIX index, continuous-time stochastic volatility models, Poisson jump, the underlying asset price process

目 录

摘要 …………………………………………………………………………………………Ⅰ

Abstract …………………………………………………………………………………… Ⅱ

1. 绪论 ………………………………………………………………………………1

1.1 引言 ………………………………………………………………………………1

1.1.1 关于期权 …………………………………………………………………1

1.1.2 VIX指数 …………………………………………………………………3

1.2经典的B-S模型 …………………………………………………………………3

1.2.1 对冲思想 …………………………………………………………………3

1.2.2 基本假设及模型形式……………………………………………………3

1.2.3 B-S模型的缺陷 …………………………………………………………5

1.3连续时间随机波动率模型…………………………………………………………6

1.4 本文的研究目的和主要研究内容 ………………………………………………7

1. 随机过程与伊藤公式 ………………………………………………………………7

2.1 维纳过程 …………………………………………………………………………7

2.1.1 定义 ………………………………………………………………………7

2.1.2 性质与特点………………………………………………………………8

2.2 二次变差定理与Ito公式………………………………………………………8

2.2.1 二次变差定理……………………………………………………………8

2.2.2 Ito积分 …………………………………………………………………9

2.2.3 Ito公式 …………………………………………………………………9

1. 3/2模型下的看涨期权表达式 ……………………………………………………10

3.1 看涨期权PDE的确定 …………………………………………………………11

• 1. 理论推导 …………………………………………………………………………12

1. 带跳的3/2模型下的看涨期权表达式 ……………………………………………14

4.1 相关理论基础 ……………………………………………………………………14

4.1.1 泊松点过程 ……………………………………………………………14

4.1.2 鞅表示……………………………………………………………………15

4.1.3 含泊松跳的伊藤公式……………………………………………………16

4.2 看涨期权PDE的确定……………………………………………………………17

1. 基础资产价格过程带跳的看涨期权表达式 ………………………………………19
   1. 二维情形下的Ito公式 …………………………………………………………19

5.2 看涨期权PDE的确定 ……………………………………………………………20

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结论 ………………………………………………………………………………………26

参考文献……………………………………………………………………………………28

致谢 ………………………………………………………………………………………29

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