基于黎曼几何的套利分析研究

发布时间：2018-10-21 17:41

【摘要】：现代金融理论的一系列重大突破和创新发展,得益于H.Markowitz的组合选择理论与F.Modigliani和M.Miller的MM理论所蕴含的无套利分析均衡思想的完美结合。无套利原理既是对金融市场均衡特征的本性刻画,表现为均衡价格与投资者风险偏好的无关性,也是基于对Arrow-Debreu模型的传承,体现了资产定价基本定理的正确与合规。无套利分析堪称是现代金融数学研究的基石。无套利分析在当今金融研究中可以说无处不在,并成为几乎每个金融研究领域中的研究重点。 S. Farinelli (2009)利用黎曼几何基本理论和方法,构建并定义了基于现金流的联络形式和曲率方程,研究了金融市场的无套利特征,给出了一系列有经济学理论意义和实用价值的金融市场无套利特征的刻画。此类研究虽刚刚开始,但因其与黎曼几何完美结合,己引起众多数学与金融学专家的广泛兴趣。受到S. Farinelli (2009)等研究文稿的启发,本文将黎曼几何基本理论和方法巧妙植入到金融市场中,特别是,将金融市场看成是一个由测量(Gauge)构成的主纤维丛,找到最具有普遍性的套利度量,基于黎曼几何的理论,说明套利度量就是测量的联络。首先,定义并研究基于投资策略意义下的联络,构建对应的曲率方程,研究基于投资策略意义下的市场无套利特征；其次,结合S. Farinelli (2009)等已有研究结果,讨论基于策略与资产共同作用并产生收益情形下的联络所对应的市场无套利特性,给出了金融市场无套利的几个等价条件；最后,研究并给出了资产价格含跳跃特性的无套利特征。作为应用,给出了一个实例,说明几何无套利的本质就是远期汇率定价。
[Abstract]:A series of major breakthroughs and innovative developments in modern financial theory have benefited from the perfect combination of the combination choice theory of H.Markowitz and the theory of MM of F.Modigliani and M.Miller. The principle of no arbitrage is not only the natural description of the equilibrium characteristics of financial markets, but also the inheritance of the Arrow-Debreu model, which reflects the correctness and compliance of the basic asset pricing theorem. No arbitrage analysis is the cornerstone of modern financial mathematics research. The no-arbitrage analysis can be said to be ubiquitous in today's financial research, and has become the focus of almost every financial research field,. S. Farinelli (2009) using Riemann geometric basic theory and method. This paper constructs and defines the contact form and curvature equation based on cash flow, studies the characteristics of non-arbitrage in financial markets, and gives a series of characterizations of non-arbitrage characteristics of financial markets with economic theoretical significance and practical value. Although this kind of research is just beginning, it has attracted extensive interest of many mathematics and finance experts because of its perfect combination with Riemann geometry. Inspired by S. Farinelli (2009 and other research papers, the basic theory and method of Riemann geometry are subtly implanted into the financial market. In particular, the financial market is regarded as a main fiber bundle composed of measured (Gauge). Based on the theory of Riemann geometry, we find the most universal arbitrage metric, and prove that arbitrage metric is the connection of measurement. Firstly, we define and study the relationship in the sense of investment strategy, construct the corresponding curvature equation, and study the characteristics of market no-arbitrage in the sense of investment strategy. Secondly, combined with the existing research results, such as S. Farinelli (2009, etc. This paper discusses the characteristics of market no-arbitrage based on the interaction of strategy and assets and the relationship under the condition of generating income, and gives some equivalent conditions of non-arbitrage in financial market. The arbitrage-free property of asset price with jump property is studied and given. As an application, an example is given to show that the essence of geometric arbitrage is forward exchange rate pricing.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：O186.12;F830.9

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