O-U过程下带违约风险的最优投资问题研究

发布时间：2018-06-27 12:30

本文选题：违约风险 + O-U过程　；参考：《上海师范大学》2014年硕士论文

【摘要】：在金融数学中,用随机控制理论研究最优投资问题是一个重要的研究领域。随着全球经济的发展,投资者及投资机构几乎每天都面临着投资决策问题,研究各类模型下的最优投资问题变得尤为重要.此外,纵观当今国际金融市场,各种金融产品错综复杂,金融市场中的违约风险越来越凸显并被人们所重视.在这样的大背景下,研究带违约风险的最优投资问题不仅具有重要的理论意义,而且也具有较强的现实意义.本文选取O-U(OrnsteinΚ Uhlenbeck)过程来描述股票价格的波动,原因在于O-U过程比几何布朗运动更符合实际金融市场.利用最优控制理论和粘性解理论,研究了O-U过程下带违约风险的最优投资问题、O-U过程下带违约传染的最优投资问题、O-U过程下可违约债券的效用无差别定价问题.本文共分为六章：第一章,阐述了带违约风险的最优投资问题的选题背景、研究意义、国内外发展现状以及本文的主要研究内容. 第二章,简单介绍了本文所涉及的基础知识：O-U过程、违约过程、最优控制与粘性解理论、效用无差别定价等基本定义与定理. 第三章,假设投资者选择的投资产品为：银行存款,O-U过程下的股票以及可违约债券,建立了可违约条件下的最优投资问题的模型.以随机最优控制理论为基础,选取合适的目标函数,利用动态规划原理,推导出值函数所满足的HJB方程,证明了HJB方程粘性解的存在性.最后,利用满足正系数条件的有限差分格式进行数值计算,并对所得结果进行了分析讨论. 第四章,在第三章的基础上,假设投资者购买公司A的股票和公司B的债券,两个公司均存在违约风险,并且两公司之间具有违约传染性.通过违约强度的变化来刻画违约传染带给彼此的影响,从投资者的角度出发建立了该最优投资问题的数学模型,然后采用有限差分法进行近似求解,并对数值结果和参数进行了分析. 第五章,利用效用无差别定价方法研究了O-U过程下的可违约债券的定价问题.假设在可违约债券的有效期内投资者可以动态优化自己的投资组合,用O-U过程替代传统的几何布朗运动来刻画股价的运动轨迹,分别在投资者购买和不购买可违约债券的两种情况下,利用动态规划原理推导出值函数所满足的HJB方程,得到可违约债券的效用无差别价格,并对一些参数进行了数值分析. 第六章,总结全文并给出了可进一步研究的问题.
[Abstract]:In financial mathematics, it is an important research field to study optimal investment problem with stochastic control theory. With the development of global economy, investors and investment institutions are faced with the problem of investment decision almost every day, so it is very important to study the optimal investment problem under various models. In addition, all kinds of financial products are complicated and the risk of default is becoming more and more prominent in the international financial market. Under this background, the study of optimal investment with default risk is not only of great theoretical significance, but also of practical significance. In this paper, the O-U (Ornstein K Uhlenbeck) process is chosen to describe the volatility of stock price. The reason is that the O-U process is more consistent with the actual financial market than the geometric Brownian motion. The optimal investment problem with default risk under O-U process and the optimal investment problem with default contagion under O-U process are studied by using optimal control theory and viscous solution theory. This paper is divided into six chapters: the first chapter describes the background of the optimal investment with default risk, research significance, domestic and foreign development status and the main research content of this paper. In the second chapter, we briefly introduce the basic definitions and theorems of the basic knowledge in this paper, such as: O-U process, default process, optimal control and viscous solution theory, utility nondifferential pricing and so on. In the third chapter, assuming that the investment products chosen by investors are stocks and defaultable bonds in the process of bank deposit, the model of optimal investment problem under the condition of default is established. Based on the stochastic optimal control theory, the HJB equation satisfied by the value function is derived by selecting the appropriate objective function and using the dynamic programming principle, and the existence of the viscous solution of the HJB equation is proved. Finally, the finite difference scheme satisfying the positive coefficient condition is used to carry out the numerical calculation, and the results obtained are analyzed and discussed. In chapter 4, on the basis of chapter 3, suppose that investors buy the stock of company A and the bond of company B. both companies have default risk, and the two companies have the contagion of default. The influence of default contagion on each other is described by the variation of default intensity. The mathematical model of the optimal investment problem is established from the perspective of investors, and then the finite difference method is used to approximate solve the problem. The numerical results and parameters are analyzed. In chapter 5, the pricing problem of defaultable bonds under O-U process is studied by using utility nondifferential pricing method. Assuming that investors can dynamically optimize their portfolios during the period of validity of defaultable bonds, the O-U process is used instead of the traditional geometric Brownian motion to depict the trajectory of stock price. In the case of investors buying and not buying defaultable bonds, the HJB equation satisfied by the value function is derived by using the dynamic programming principle, and the utility of the defaultable bonds is obtained without difference, and some parameters are analyzed numerically. The sixth chapter summarizes the full text and gives the problems that can be further studied.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：F830.91;O211.6

【相似文献】