随机微分方程在金融定价中的应用

目 录

中文摘要 1

英文摘要 2

1 前言 3

1.1金融现状简介 3

1.2 本文主要工作 3

2 债券价格计算 4

3 股票价格计算及投资最优控制 6

3.1 基本定义和引理 6

3.2 股价运动的随机分析模型 7

3.3模型证明 8

3.4 2元模型的风险控制 123.4.2主要结论 12

3.4.3 应用举例 14

4期权定价 16

4.1期权概念及性质 16

4.2期权定价公式 17

4.3应用举例 20

5总结 21

参考文献 22

致谢 23

Abstract

Along with societys prosperity progress, the financial market also develops and consummates, and acts the more and more important role in the national economy and life. The operation of financial market demonstrates the enormous randomness, thus the stochastic differential equation theory has the very good opportunity in the aspect of financial product fixed price. Therefore applied the stochastic differential equation in the financial domain is a hot topic of discussion in recently several years.

The present paper carries on the discussion to this kind of application. It mainly carries on the continuous process stochastic differential equation discretization of the research. The concrete use mathematics modelling method, and take the stochastic differential equation related theory and the money market theory as the main tool to study the financial market negotiable securities, the stock as well as the stock time power fixed price and get the correlation content and the corresponding conclusion as well as the bond, the stock, the stock time power price computation expression, and conformed to the geometry Brown movement model to the stock stochastic price model to produce the detailed rigorous mathematic proof process. Finally this paper has studied this research area extant question, and has forecasted this domain research development direction.

Keywords：stochastic differential equation; option pricing; wiener’s process; optimal-investment

摘 要

随着社会的繁荣进步,金融市场也不断发展和完善，并在国民经济生活中扮演着越来越重要的角色。金融市场的运作显示出极大的随机性，因而随机微分方程理论在金融产品定价方面有很好的用武之地。所以将随机微分方程应用于金融领域是最近几年的1个热门话题。

本论文即是对这种应用进行探讨。技术上的思想主要是将连续过程的随机微分方程离散化来进行研究。具体使用数学建模的方法，以随机微分方程的有关理论和金融市场理论为主要工具研究了金融市场有价证券、股票以及股票期权的定价及相关内容，得出了相应结论以及债券，股票，股票期权价格的计算表达式，并对股票的随机价格模型符合 几何布朗运动模型给出了详细严谨的数学证明过程。最后分析了本研究领域现存的问题，预测了本领域研究的发展方向。

关键词： 随机微分方程；期权定价；维纳过程；最优投资