分数布朗运动环境下的欧式与美式期权定价研究

发布时间：2018-04-28 04:22

本文选题：分数布朗运动 + 拟条件数学期望　；参考：《宁夏大学》2013年硕士论文

【摘要】：金融衍生品作为一种金融创新工具在国际金融市场上起着日益重要的作用.作为其四大门类之一的期权,更是因其能够通过组合的形式复制其他金融衍生品而备受关注.期权的定价问题一直是现代金融领域研究的核心.考虑到现实金融产品所处环境的复杂性,各种期权的定价研究近年来已成为期权研究领域的热门课题. 自从B-S期权定价模型问世以来,金融界对金融衍生产品的定价问题越来越重视,.在各种不同的假设条件下,不断对模型进行改进,最终证实股票的市场价格不是简单应用原始的B-S定价公式就能描述的,应是一个具有长期依赖性和自相似性的,资本市场也是持久性的时间序列.这就要求应用一个具有长期记忆的过程来描述市场的结构特性.而引入分数布朗运动作为随机变量可以更加准确地刻画金融市场的波动,更符合实际情况.本文主要讨论在分数布朗运动环境下的欧式和美式期权的定价研究.接着引入了混合分数布朗运动,并给出在混合分数布朗运动环境下的欧式及美式期权的定价公式. 第一章,介绍了分数布朗运动环境下期权研究的背景意义.早期的期权定价理论介绍,在提出经典B-S期权定价模型之后,期权定价问题的研究及发展,以及本文主要内容的介绍. 第二章,相关基础知识介绍：随机过程及相关鞅理论,应用鞅变换理论得到拟条件数学期望；引出分数布朗运动,并应用分数型风险中性测度得到期权的价格. 第三章,应用拟条件数学期望推导出分数布朗运动环境下欧式双向期权的定价公式及两种资产和多资产的最大值期权公式,并拓展到由多维分数布朗运动与几何布朗运动的线性组合构成的混合分数布朗运动下最大值期权定价公式,进而又讨论分析了股价模型中所涉及的五种避险参数对期权价格的影响。第四章,应用数值法求解出分数布朗运动环境下的金融衍生品满足的统一的偏微分方程,得到带有红利的美式期权的定价公式并给出混合分数布朗运动环境下的美式期权定价公式. 第五章,总结本文得出的所有结论,并得出本文相应问题在今后应注意改进的方面.
[Abstract]:As a financial innovation tool, financial derivatives play an increasingly important role in the international financial market. As one of its four categories, option has attracted much attention because of its ability to replicate other financial derivatives in the form of combination. Option pricing has always been the core of modern financial research. Considering the complexity of the environment in which the real financial products are located, the research on the pricing of various options has become a hot topic in the field of options research in recent years. Since the emergence of B-S option pricing model, financial circles have paid more and more attention to the pricing of financial derivatives. Under various hypothetical conditions, the model is continuously improved, and finally it is proved that the market price of stock is not simply described by the original B-S pricing formula, but should be a long-term dependent and self-similar one. Capital markets are also a persistent time series. This requires the application of a long-term memory process to describe the structural characteristics of the market. The introduction of fractional Brownian motion as a random variable can more accurately describe the volatility of financial markets, and more in line with the actual situation. This paper mainly discusses the pricing of European and American options in fractional Brownian motion. Then the mixed fractional Brownian motion is introduced, and the pricing formulas of European and American options under the mixed fractional Brownian motion environment are given. In the first chapter, the background significance of option research in fractional Brownian motion is introduced. After the introduction of the classical B-S option pricing model, the research and development of the option pricing problem and the main content of this paper are introduced. In the second chapter, the basic knowledge is introduced: stochastic process and martingale theory, applying martingale transformation theory to obtain quasi conditional mathematical expectation, leading to fractional Brownian motion, and applying fractional risk neutral measure to obtain the price of option. In chapter 3, the pricing formula of European two-way option and the maximum option formula of two kinds of assets and multiple assets under fractional Brownian motion environment are derived by using quasi conditional mathematical expectation. And the maximum option pricing formula under mixed fractional Brownian motion is extended to a mixed fractional Brownian motion composed of linear combination of multidimensional fractional Brownian motion and geometric Brownian motion. Furthermore, the influence of five hedge parameters involved in the stock price model on the option price is discussed. In chapter 4, the unified partial differential equation of financial derivatives under fractional Brownian motion is solved by numerical method. The pricing formula of American option with dividend is obtained and the pricing formula of American option under mixed fractional Brownian motion is given. The fifth chapter summarizes all the conclusions of this paper and concludes that the corresponding problems in this paper should be improved in the future.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F830.91;F224;O211.6

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