分数布朗运动环境下的美式期权定价

发布时间：2018-04-09 04:38

本文选题：分数布朗运动　切入点：复合期权　出处：《华中师范大学》2013年硕士论文

【摘要】：美式期权作为一种重要的期权,它的定价问题越来越受到人们的关注。现存的美式期权的定价方法大多数是以标的资产服从几何布朗运动为前提的。然而大量的实证分析表明：标的资产一般具有长期依赖性和自相关性。这与几何布朗运动的特性不相符合,而恰恰与分数几何布朗运动的特性相吻合。因此为了使研究结果更接近现实,这篇文章主要研究了分数布朗运动环境下的美式期权定价。本文在第二部分首先介绍了复合期权的的相关概念,通过对执行时刻进行深入分析和讨论,运用测度变换技巧及拟鞅定价方法,得到了分数布朗运动环境下复合期权的解析解。其次,讨论了带成比例交易费用的美式期权的定价问题。通过将执行时刻限制在当前和到期日之间的一些事先确定的时刻,就可以把美式期权看作一系列的复合期权,从而应用复合期权近似得到了美式看跌期权的近似公式。为了使所建立的模型更加接近金融市场的实际情况。本文在第三部分考虑了在期权有效期内标的资产有红利支付的美式期权的定价问题,运用分数布朗运动随机积分理论及拟鞅定价方法,得到了单次红利支付的美式看涨期权的解析解。并对看跌期权进行了深入的讨论和研究,但是,由于其最佳执行策略的复杂性,单次红利支付的美式看跌期权不存在解析解。
[Abstract]:American option as an important option, its pricing problem has been paid more and more attention.Most existing American option pricing methods are based on geometric Brownian motion of underlying assets.However, a large number of empirical analysis shows that the underlying assets generally have long-term dependence and self-correlation.This does not accord with the characteristic of geometric Brownian motion, but with the characteristic of fractional geometric Brownian motion.Therefore, in order to make the results closer to reality, this paper mainly studies the pricing of American options in fractional Brownian motion environment.In the second part of this paper, we first introduce the concept of compound option. By analyzing and discussing the execution time, using measure transformation technique and quasi martingale pricing method, we obtain the analytic solution of composite option in fractional Brownian motion environment.Secondly, the pricing of American options with proportional transaction costs is discussed.By limiting the execution time to some predetermined moments between the current and due dates, the American option can be regarded as a series of composite options, and the approximate formula of the American put option is obtained by applying the compound option approximation.In order to make the model closer to the actual situation of the financial market.In the third part of this paper, we consider the pricing problem of the American option with dividend payment of the underlying asset during the period of validity of the option, and apply the stochastic integral theory of fractional Brownian motion and the quasi-martingale pricing method.The analytic solution of American call option with single dividend payment is obtained.Put options are discussed and studied deeply. However, due to the complexity of the optimal execution strategy, there is no analytical solution for the American put options with single dividend payment.
【学位授予单位】：华中师范大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F830.9;F224;O211.6

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