随机环境中服从分数布朗运动的期权研究

发布时间：2018-04-24 05:32

本文选题：分数布朗运动 + 随机利率　；参考：《上海交通大学》2013年硕士论文

【摘要】：数学理论在金融领域中应用的一个关键问题是要解决风险资产及其衍生品的定价问题，而期权定价是其中的一个重要内容，在经典的理论中通常使用布朗运动来刻画资产价格的变化。与布朗运动相比，分数布朗运动具有增量间相关的性质，用它来刻画资产价格能够更好的体现出非成熟金融市场中股票收益率分布呈现的尖峰、厚尾等特征。它的研究对于金融数学和资本市场都有广泛而深远的影响。本文综述了已有的有关分数布朗运动的国内外研究成果，并在前人的基础上引入了更多的随机假设，，求出了相关期权的定价公式，获得的主要结果包括：1.将扩展的Vasick利率模型引入到分数指数O U过程中，分别对标的资产价格服从分数指数O U过程中欧式期权和重置期权的定价进行了研究，得出了相应的定价公式；2.在更一般的随机环境中讨论了服从分数布朗运动的欧式看涨期权定价问题，用等价鞅测度的方法，分别研究了随机的期望收益率和利率、随机的执行价格下欧式看涨期权，并给出了定价公式的具体表达形式。
[Abstract]:One of the key problems in the application of mathematical theory in the field of finance is to solve the pricing problem of risky assets and their derivatives, and option pricing is one of the important contents. In classical theory, Brownian motion is usually used to describe the change of asset price. Compared with the Brownian motion, fractional Brownian motion has the property of increment correlation, which can describe the asset price better to reflect the peak and thick tail of the stock yield distribution in the immature financial market. Its research has extensive and profound influence on financial mathematics and capital market. In this paper, the domestic and foreign research results on fractional Brownian motion are reviewed, and more random assumptions are introduced on the basis of previous studies, and the pricing formulas of relevant options are obtained. The main results obtained include: 1. The extended Vasick interest rate model is introduced into the process of fractional exponent O U, and the pricing of European option and replacement option in the process of fractional index O U is studied respectively, and the corresponding pricing formula is obtained. In a more general stochastic environment, the pricing problem of European call options with fractional Brownian motion is discussed. By using the method of equivalent martingale measure, the stochastic expected return rate and interest rate, and the European call options at random executive price are studied respectively. The expression of pricing formula is given.
【学位授予单位】：上海交通大学
【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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