鞅方法和随机控制理论在投资组合和期权定价中的应用

发布时间：2018-04-28 14:45

本文选题：鞅 + 随机控制　；参考：《中国科学技术大学》2012年博士论文

【摘要】：投资组合选择和期权定价是现代数理金融理论的两大研究主题,经典投资组合选择问题的研究通常建立在Markowitz均值-方差或von Neumann-Morgenstern期望效用框架下,探讨风险厌恶投资者的理性最优投资行为；而经典期权定价问题的研究则是利用无套利定价原理,探讨合理的期权价格. 本文将利用鞅方法和随机控制理论从不同的角度对这两个问题展开讨论,一方面在非von Neumann-Morgenstern期望效用理论框架下,研究“非理性”投资者的最优投资行为；另一方面将投资问题与期权定价问题建立在一个模型框架下.同时分析投资者的最优股票、期权投资以及期权定价问题. 首先,系统研究了损失厌恶投资者的最优投资组合选择模型,其中,在连续时间完全市场框架下,分别考虑财富非负和带有基准下限约束的情形,讨论了一般价值函数下的损失厌恶投资者的最优投资问题,分析了损失厌恶投资者的投资组合保险策略,讨论了解的存在唯一性,并和经典期望效用最大化意义下的结果做了对比分析；另外,对于不完全市场以及跳-扩散市场情形下的最优投资问题,也做了细致的分析. 其次,在秩相依期望效用理论框架下,研究了带有财富VaR约束的最优投资组合选择模型,利用分位数函数技术对模型进行求解,获得了最优期末财富,并分析了一个例子,得到了最优财富过程及最优资产配置策略. 最后,研究了标的资产为不可交易资产的期权定价以及最优股票、期权投资问题.在风险偏好是指数形式的假定下,建立了股票和期权最优配置之间的动态关系,获得了动态期权定价公式.特别地,利用Feynman-Kac公式,给出了封闭形式的价格表示；同时,分析了解的性质,讨论了动态均衡价格与边际价格和效用无差异价格之间的关系.
[Abstract]:Portfolio selection and option pricing are two major research topics in modern mathematical finance theory. The classical portfolio selection problem is usually based on the framework of Markowitz mean-variance or von Neumann-Morgenstern expected utility. The rational optimal investment behavior of risk-averse investors is discussed, while the classical option pricing problem is studied by using the no-arbitrage pricing principle to explore the reasonable option price. In this paper, we discuss these two problems from different angles by using martingale method and stochastic control theory. On the one hand, we study the optimal investment behavior of "irrational" investors under the framework of non- Neumann-Morgenstern expected utility theory. On the other hand, the investment problem and option pricing problem are established in a model framework. At the same time, it analyzes the optimal stock, option investment and option pricing of investors. First of all, the optimal portfolio selection model for loss-averse investors is studied systematically. Under the framework of complete market with continuous time, the non-negative wealth and the lower bound of benchmark are considered, respectively. This paper discusses the optimal investment of loss averse investors under the general value function, analyzes the portfolio insurance strategy of loss averse investors, and discusses the existence and uniqueness of the solution. In addition, the optimal investment problem in incomplete market and jump-diffusion market is analyzed in detail. Secondly, under the framework of rank dependent expected utility theory, the optimal portfolio selection model with wealth VaR constraints is studied. The optimal final wealth is obtained by using quantile function technique, and an example is given. The optimal wealth process and the optimal asset allocation strategy are obtained. Finally, the paper studies the option pricing of the underlying asset as a non-tradable asset and the problem of optimal stock and option investment. Under the assumption that risk preference is an exponential form, the dynamic relationship between stock and option optimal allocation is established, and the dynamic option pricing formula is obtained. In particular, by using the Feynman-Kac formula, the closed form of price representation is given, and the properties of the solution are analyzed, and the relationship between the dynamic equilibrium price and marginal price and utility price is discussed.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2012
【分类号】：F224;F830.59

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