基于分数布朗运动环境下期权定价的若干问题研究 Alternative Title The Research of Several Problems about Pricing Options 冯杰才 Thesis Advisor 严定琪 2009-05-28 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 分数布朗运动 期权定价 时变参数 欧式期权 美式期权 Abstract 期权定价问题是金融工程中一个重要的问题，多年来诸多学者对此问题进行了研究。其中最为著名的是1973 年Black ¡ Scholes 公式的发现，此公式也成为日后学者们研究的基础。Black ¡ Scholes 公式有一个很重要的前提条件：原生资产价格演化遵循布朗运动。但是，随着近年来学者们研究发现，证券市场的运行并不遵循布朗运动，而是服从更为一般的分数布朗运动。因此，以更为一般的分数布朗运动代替标准布朗运动来进行期权定价问题的研究以成为当前研究的一个主要方向。 本文首先介绍了分数布朗运动并对分数布朗运动下Black Scholes 公式进行了推导。然后通过研究美式期权的性质，给出了分数布朗运动环境下带红利的永久美式期权的定价。研究了分数布朗运动下时变参数的欧式期权定价，通过函数变换将时变参数问题转化为非时变参数的问题，并最终得出了欧式期权定价公式。最后通过引用文献[21]中实证分析的结果来说明中国股票市场不服从随机游走模型，而是表现出有偏随机游动的特性，具有长期记忆性。这就是本论文的出发点：标的资产服从分数布朗运动。 Other Abstract The problem of options pricing is an important scope of financial engineering,many scholars have researched this problem in these years. The most famous work is the found of Black-Scholes formula in 1973, which is also became the foundation of research work in the following years. The Black-Scholes formula has an important assumption: the evolution of underlying asset follows the Brownian motion. However, due to these years' study, the security market run following the general fractional Brownian motion instead of the Brownian motion. Therefore,using the fractional Brownian motion to find the options' price is a major research field nowadays. In this paper, it introduced the fractional Brownian motion firstly, and derived the Black-Scholes formula under the fractional Brownian motion. It developed the pricing of perpetual American option with dividend under the fractional Brownian motion, through the study of American option's properties. It gave the European option pricing with Time-varying parameters under the fractional Brownian motion. Through function transformation, it changed this problem to the non Time-varying parameters problem, and it got the European option pricing at the end. Finally, through quoted the empirical analysis result of document[21],it showed the Chinese stock market doesn't follow the random walk model, besides, it showed the biased random walk and Long-term memory characteristics.This is the starting point for this paper: the underlying asset follows the fractional Brownian motion. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225178 Collection 数学与统计学院 Recommended CitationGB/T 7714 冯杰才. 基于分数布朗运动环境下期权定价的若干问题研究[D]. 兰州. 兰州大学,2009.
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