分数布朗运动环境中欧式新型期权的定价

发布时间：2018-02-17 01:23

本文关键词： 分数布朗运动幂型期权跳—扩散过程拟鞅定价亚式期权　出处：《东北大学》2013年硕士论文　论文类型：学位论文

【摘要】：随着金融市场的不断发展与壮大,对金融衍生产品的定价,特别是对新型期权的定价问题是金融学领域的主要研究内容,更是金融数学研究领域的主要内容之一。经典的B-S期权定价模型假设标的资产服从几何布朗运动,但是通过大量的实验证明,标的资产的价格运动特征与分数布朗运动的特征相符合。因此本文主要引入分数布朗运动,假设标的资产的价格服从分数布朗运动过程,研究了两种新型期权的定价模型。本文的主要研究工作分为以下三个部分：第一部分：利用时间轴变换法的思想,将几何分数布朗运动近似地转化为几何布朗运动,建立了分数欧式幂型期权的定价模型,其中包括有红利支付的分数欧式看涨期权,并且得到分数欧式幂型看涨期权与看跌期权的平价关系。第二部分：规定扩散过程服从分数布朗运动,跳跃过程为计数过程,利用分数随机分析理论中的拟鞅定价方法及风险中性定价理论,推导出分数跳—扩散欧式幂型期权定价的解析表达式。第三部分：利用投资组合及对冲原理,得到了分数布朗运动环境中亚式期权满足的偏微分方程,再通过转换变形,将方程转化为满足柯西问题的一维偏微分方程,建立具有固定执行价格的分数几何平均亚式看涨期权的定价模型。
[Abstract]:With the development and expansion of financial market, the pricing of financial derivatives, especially the pricing of new options, is the main research content in the field of finance. The classical B-S option pricing model assumes that the underlying asset is moving from geometric Brownian motion, but it is proved by a large number of experiments. Therefore, this paper mainly introduces fractional Brownian motion. Assuming that the price of underlying assets is based on fractional Brownian motion, two new pricing models of options are studied. The main research work of this paper is divided into the following three parts:. Part one: using the idea of time-axis transformation, the geometric fractional Brownian motion is transformed into geometric Brownian motion approximately, and the pricing model of fractional European power-type option is established, which includes fractional European call option with dividend payment. The parity relationship between fractional European power call option and put option is obtained. In the second part, we use the quasi-martingale pricing method and risk-neutral pricing theory in fractional stochastic analysis theory to specify diffusion process from fractional Brownian motion and jump process as counting process. The analytical expression of fractional hopping-diffusion European power option pricing is derived. In the third part, by using the portfolio and hedging principle, the partial differential equation of the Central Asian option in the fractional Brownian motion environment is obtained, and then the equation is transformed into a one-dimensional partial differential equation satisfying the Cauchy problem by transforming the equation into a one-dimensional partial differential equation which satisfies the Cauchy problem. The pricing model of fractional geometric average call option with fixed execution price is established.
【学位授予单位】：东北大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F830.91;O211.6

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