随机利率下的期权定价模型

发布时间：2018-01-08 18:29

本文关键词：随机利率下的期权定价模型　出处：《西安工程大学》2013年硕士论文　论文类型：学位论文

【摘要】：期权定价理论是金融数学研究的核心问题之一。1973年Black和Scholes假设股票价格在几何布朗运动环境下，提出了著名的Black-Scholes期权定价模型，并给出了其定价公式。然而，在金融市场模型中用分数布朗运动取代标准布朗运动早已被众多学者认同，主要是由于分数布朗运动具有较好地“厚尾”和长程依赖性。在实际的金融市场中，利率是随机的。同时本文假设股票价格满足分数布朗运动驱动的随机微分方程，建立随机分数布朗运动环境下具有随机利率的金融市场数学模型，研究缺口期权和两值期权定价问题。全文共分五章。第一章，介绍期权定价理论的历史及研究现状、选题依据以及研究的主要内容。第二章，主要介绍分数布朗运动的定义和性质；同时也介绍了欧式期权保险精算方法。第三章，假设股票价格满足分数布朗运动驱动的随机微分方程，利率满足Vasicek模型，建立金融市场数学模型，利用保险精算方法和分数布朗运动的随机分析理论，得到缺口期权的定价公式。第四章，在分数布朗运动环境下具有随机利率的金融市场数学模型中，利用保险精算方法和分数布朗运动的随机分析理论讨论了两值期权的定价问题，，得到两值期权的定价公式。第五章，总结本文主要结果，并提出进一步研究问题。
[Abstract]:Option pricing theory is one of the core problems in financial mathematics. In 1973, Black and Scholes assumed that the stock price is in the geometric Brownian motion environment. In this paper, a famous Black-Scholes option pricing model is proposed and its pricing formula is given. Replacing standard Brownian motion with fractional Brownian motion in financial market model has been accepted by many scholars, mainly because fractional Brownian motion has good "thick tail" and long range dependence. At the same time, this paper assumes that the stock price satisfies the stochastic differential equation driven by fractional Brownian motion, and establishes a mathematical model of financial market with stochastic interest rate in the environment of stochastic fractional Brownian motion. This paper studies the pricing of gap options and two-valued options. The first chapter introduces the history and research status of option pricing theory. In the second chapter, the definition and properties of fractional Brownian motion are introduced, and the actuarial method of European option insurance is also introduced. In the third chapter, assuming that the stock price satisfies the stochastic differential equation driven by fractional Brownian motion and the interest rate satisfies the Vasicek model, the mathematical model of financial market is established. By using the actuarial method and the stochastic analysis theory of fractional Brownian motion, the pricing formula of gap option is obtained. In Chapter 4th, in the financial market mathematical model with stochastic interest rate in the environment of fractional Brownian motion, the pricing problem of two-valued options is discussed by using the actuarial method of insurance and the stochastic analysis theory of fractional Brownian motion. The pricing formula of two-valued option is obtained. Chapter 5th summarizes the main results of this paper and puts forward further research problems.
【学位授予单位】：西安工程大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F830.9;F224

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