求解美式期权定价的高阶精度紧致有限差分法

发布时间：2018-04-11 09:53

本文选题：美式期权 + 体制转换　；参考：《闽南师范大学》2017年硕士论文

【摘要】：期权定价理论作为金融领域中最重要的发展之一,是目前金融数学研究的重点问题。期权价格作为影响买卖双方收益的直接因素,成为期权定价理论的重点研究对象。期权分为欧式期权和美式期权,与欧式期权相比,美式期权要比欧式期权复杂很多。其中,美式期权的研究也因此成为期权定价理论的核心问题。近年来金融的很多领域应用体制转换模型,并且在对相关的金融数学研究时应用体制转换模型取得了更好的结果。在前人研究的基础上,本文着重讨论求解基于体制转换模型的美式期权定价问题的一种高精度紧致有限差分格式。首先,本文考虑基于Black-Scholes模型的美式期权定价问题,先对Black-Scholes方程做代数变换,消除方程中对空间上的一阶导数,得到美式期权定价问题的数学模型,进而提出高精度的紧致有限差分格式。然后,应用离散能量法分析了该格式的稳定性和收敛性。最后,给出两个数值算例,其数值结果证实了理论分析及该格式的应用可行性。其次,本文考虑基于体制转换模型的美式期权定价问题,该问题满足由M个自由边界值问题组成的方程组,这使得问题很难得到解决。我们先对最初体制转换下美式期权满足的方程组作代数变换,消除方程组中对空间上的一阶导数,得到体制转换模型下美式期权定价问题的数学模型,进而提出高精度的紧致有限差分格式。然后,应用离散能量法分析了该格式的稳定性和收敛性。最后,给出了几个数值算例,其数值结果证实了理论分析及该格式的应用可行性。
[Abstract]:As one of the most important developments in the field of finance, option pricing theory is one of the most important problems in financial mathematics.Option price, as a direct factor affecting the income of buyers and sellers, has become an important research object of option pricing theory.Options are divided into European options and American options. Compared with European options, American options are much more complex than European options.Among them, the research of American option becomes the core problem of option pricing theory.In recent years, the system transformation model has been applied in many fields of finance, and better results have been obtained in the study of financial mathematics.On the basis of previous studies, this paper focuses on a compact finite-difference scheme with high precision for solving American option pricing problems based on system transformation model.First of all, this paper considers the American option pricing problem based on Black-Scholes model. Firstly, the Black-Scholes equation is algebraic transformed to eliminate the first derivative in the space of the equation, and the mathematical model of American option pricing problem is obtained.Then a compact finite difference scheme with high accuracy is proposed.Then, the stability and convergence of the scheme are analyzed by using the discrete energy method.Finally, two numerical examples are given. The numerical results verify the theoretical analysis and the feasibility of the scheme.Secondly, this paper considers the American option pricing problem based on the system transformation model, which satisfies the equations composed of M free boundary value problems, which makes it difficult to solve the problem.First, we make algebraic transformation of the equations satisfied by American options under the initial system transformation, eliminate the first derivative of the equations on the space, and obtain the mathematical model of the pricing problem of American options under the system transformation model.Then a compact finite difference scheme with high accuracy is proposed.Then, the stability and convergence of the scheme are analyzed by using the discrete energy method.Finally, several numerical examples are given. The numerical results confirm the theoretical analysis and the feasibility of the scheme.
【学位授予单位】：闽南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.3;F830.93

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