鞅理论及其在某些金融模型中的应用

发布时间：2018-07-03 10:06

本文选题：鞅理论 + 期权定价　；参考：《哈尔滨工业大学》2012年硕士论文

【摘要】：鞅论是概率论中的一个重要独立分支，是概率论与随机过程等方面的基础。鞅理论具有很多的实际意义，在定价决策和控制模型中都起着重要的作用，我们可以通过鞅论框架的构造使复杂问题简单化，因此被广泛使用，同时鞅论还渗透到统计分析、调和分析、Banach空间几何学以及随机分析等方面，同时取得了丰富的成果。本文主要内容分为两部分，第一部分综述了鞅的理论，，分经典鞅论和现代鞅论两部分来介绍，经典鞅论内容主要包括离散时间参数鞅和连续时间参数鞅的定义、性质、不等式、收敛定理、停时定理等。现代鞅论内容主要包括鞅的分解、可积变差鞅、平方可积鞅、局部鞅、半鞅、鞅的极限理论和鞅观点下的随机积分问题。其中鞅理论和随机积分知识相结合形成鞅方法，在金融领域有很重要的实际意义。第二部分的内容主要是关于鞅理论的应用，我们分别引入了单期模型、多期模型和经典的Black—Scholes期权定价模型，通过鞅理论的知识对这些模型进行分析，得到相应的结果，最后建立基于指数Ornstein-Uhlenbeck过程的带有红利支付的不确定执行价格的期权模型，并分别计算出在欧式期权和美式期权的情况下期权不同的定价。可以看出鞅理论内容在金融领域期权定价方面具有很强的实用价值，随着鞅理论的不断完善给金融领域的相关衍生品定价提供了理论基础。
[Abstract]:Martingale theory is an important independent branch of probability theory and the basis of probability theory and stochastic process. Martingale theory has a lot of practical significance and plays an important role in pricing decision and control model. We can simplify complex problems by constructing martingale theory frame, so it is widely used, and martingale theory also permeates statistical analysis. Harmonic analysis of Banach space geometry and random analysis, and achieved a wealth of results. The main contents of this paper are divided into two parts. The first part summarizes the theory of martingale, which is introduced in two parts: classical martingale theory and modern martingale theory. The classical martingale theory mainly includes the definition, properties, inequalities of discrete time parameter martingale and continuous time parameter martingale. Convergence theorem, stop time theorem and so on. The content of modern martingale theory mainly includes the decomposition of martingale, integrable martingale, square integrable martingale, local martingale, semimartingale, martingale limit theory and stochastic integral problem under martingale viewpoint. Martingale theory and stochastic integral knowledge combine to form martingale method, which is of great practical significance in the field of finance. The second part is mainly about the application of martingale theory. We introduce the single-period model, multi-period model and the classic Black-Scholes option pricing model respectively. Through the knowledge of martingale theory, we analyze these models and get the corresponding results. Finally, an option model with uncertain executive price with dividend payment is established based on the exponential Ornstein-Uhlenbeck process, and the different pricing of options is calculated in the case of European option and American option respectively. It can be seen that martingale theory has a strong practical value in the field of financial option pricing. With the continuous improvement of martingale theory, it provides a theoretical basis for the pricing of related derivatives in the financial field.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：O211.6;F830.9

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