分数布朗运动下奇异期权的定价

发布时间：2018-06-16 19:12

本文选题：分数布朗运动 + wick积　；参考：《新疆大学》2012年硕士论文

【摘要】：众所周知,在20世纪随着金融业的发展,金融市场的日益完善,人们出于规避风险和投资的需求而设计了许多金融衍生工具.期权既是一种有效的避险手段,又是一种绝妙的投机方法,这使得期权倍受投资者和投机者的青睐,通过期权定价方法有可能找到一般衍生证券的定价理论. 标的资产受布朗运动驱动的期权定价问题日趋完美,但是现实的股票市场大多具有长期记忆性,这使得用布朗运动来模拟股价运动偏差较大,而分数布朗运动的性质使得它能更好的模拟股价波动.由于理论的局限性,分形市场下期权定价问题一度进展缓慢,近年来受理论的发展,分形市场下期权定价问题快速发展. 本文利用分数风险中性定价原理、分数Girsanov定理、拟鞅和测度变换等方法,讨论了标的资产服从分数布朗运动的远期开始期权和波士顿期权的定价问题,以及讨论了标的资产服从多个分数布朗运动的领子期权和再装期权的定价问题. 本文的主要内容共分为四章：第一章为绪论部分,简单的介绍了期权,期权定价的发展路程和分数布朗运动下期权定价研究的基本情况,最后说明了本文的主要内容. 第二章为预备知识,介绍了分数布朗运动,wick积分,拟条件期望,拟鞅以及其相关性质,最后引入几个相关的引理. 第三章,在股价受单个分数布朗运动驱动的市场模型下,主要讨论了标的资产服从单个分数布朗运动的远期开始期权和波士顿期权的定价,利用测度变换和拟鞅方法,得到了其一般形式的定价公式. 第四章,在股价受多个分数布朗运动驱动的市场模型下,主要讨论了标的资产服从多个分数布朗运动领子期权和再装期权的定价问题,利用测度变换和拟鞅方法,得到了其一般形式的定价公式.
[Abstract]:As we all know, with the development of financial industry and the perfection of financial market in the 20th century, many financial derivatives have been designed in order to avoid risk and investment. Option is not only an effective means of avoiding risk, but also a brilliant method of speculation, which makes the option favored by investors and speculators. Through the option pricing method, it is possible to find the pricing theory of general derivative securities. The option pricing problem of underlying assets driven by Brownian motion is becoming more and more perfect, but most of the real stock market has long-term memory, which makes the error of using Brownian motion to simulate stock price movement is large. And the property of fractional Brownian motion makes it can better simulate stock price fluctuation. Due to the limitation of the theory, the option pricing problem in fractal market has made a slow progress. In recent years, the option pricing problem in fractal market has developed rapidly due to the development of theory. In this paper, by using the fractional risk neutral pricing principle, fractional Girsanov theorem, quasi martingale and measure transformation, we discuss the pricing of forward starting options and Boston options for the underlying asset from fractional Brownian motion. The pricing of collar options and reload options for underlying asset garments from multiple fractional Brownian motions is also discussed. The main contents of this paper are divided into four chapters: the first chapter is the introduction, which introduces the options, the development of option pricing and the basic situation of option pricing under fractional Brownian motion. Finally, the main contents of this paper are explained. In the second chapter, we introduce the fractional Brownian motion, quasi conditional expectation, quasi martingale and its related properties. Finally, we introduce some relevant lemmas. In chapter 3, under the market model of stock price driven by single fractional Brownian motion, we mainly discuss the pricing of forward start option and Boston option of underlying asset from single fractional Brownian motion, using measure transformation and quasi martingale method. The general pricing formula is obtained. In chapter 4, under the market model driven by multiple fractional Brownian motion, we mainly discuss the pricing problem of the underlying asset from multiple fractional Brownian motion collar options and reload options, and use the measure transformation and quasi-martingale method. The general pricing formula is obtained.
【学位授予单位】：新疆大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F224;F830.9

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