美式期权定价模型数值方法

发布时间：2018-02-28 15:01

本文关键词： 美式期权定价自由边界问题 Front-fixing方法三层格式紧致差分格式　出处：《中国海洋大学》2013年硕士论文　论文类型：学位论文

【摘要】：期权定价理论的创建是近几十年金融领域中最重要的发展之一。期权价格直接影响到买卖双方的盈亏状况，是现代期权理论研究重点方向。相对于欧式期权，美式期权可以提前实施，拥有更多的获利机会，操作具有更大的灵活性，应用更为广泛。因此美式期权定价问题成为期权定价理论的核心问题，对美式期权定价模型的研究也就更具有实际意义。现代期权定价理论的基础是Black-Scholes定价模型的建立。Black-Scholes定价模型在有效市场和股票价格遵循几何布朗运动等一系列假设条件下，利用对冲技巧，得出的欧式期权Black-Scholes定价模型。本文进一步假设市场风险是中性的，修正Black-Scholes方程的分析过程，将期权定价模型推广支付红利的美式期权情况下。在一般情况下，美式期权定价模型没有解析的定价公式，本文研究支付红利的美式期权定价Black-Scholes方程的几种数值解法。本文组织如下，，引言部分，简要介绍了金融衍生工具以及期权的知识，叙述了国内外对美式期权定价模型数值解法的研究现状。第二部分详细地论述了欧式期权Black-Scholes定价模型的建立过程和最优执行边界。本文第三部分，通过对支付红利的美式买入期权Black-Scholes方程进行Front-fxing变量替换，将自由边界问题转化为一个参数非线性的定边界问题，并利用美式期权自由边界性质，将空间区域限定在一个矩形区域上。对此非线性问题构造三层的二阶差分格式，并使用两层格式预测矫正技术计算第一层，最后进行数值实验，与二叉树方法进行比较证明算法是有效的。第四部分对非线性问题，构造三层的紧致差分格式，第一层的计算使用预测矫正技术，进行数值实验并与二阶差分方法对比，证明紧致差分算法比二阶差分算法精度高。本文的最后给出了相关的结论。
[Abstract]:The establishment of option pricing theory is one of the most important developments in the field of finance in recent decades. American options can be implemented in advance, have more profit opportunities, have greater flexibility in operation, and are more widely used. Therefore, American option pricing problem has become the core issue of option pricing theory. The study of American option pricing model is of practical significance. The foundation of modern option pricing theory is the establishment of Black-Scholes pricing model. Black-Scholes pricing model utilizes hedging techniques under a series of assumptions such as efficient market and stock price following geometric Brownian motion. This paper further assumes that the market risk is neutral, modifies the analysis process of Black-Scholes equation, and generalizes the option pricing model in the case of American option which pays dividends. In general, there is no analytic pricing formula for American option pricing model. In this paper, several numerical solutions to the Black-Scholes equation of American option pricing for dividend payment are studied. This paper is organized as follows: introduction, briefly introduces the knowledge of financial derivatives and options, This paper describes the research status of numerical solution to American option pricing model at home and abroad. The second part discusses the process of establishing European option Black-Scholes pricing model and the optimal execution boundary in detail. The third part of this paper, By replacing the Black-Scholes equation of American call option with dividend, the free boundary problem is transformed into a fixed boundary problem with nonlinear parameters, and the free boundary property of American option is used. The spatial region is limited to a rectangular region. A three-layer second-order difference scheme is constructed for this nonlinear problem, and the first layer is calculated by using the two-layer scheme to predict the correction technique. Finally, numerical experiments are carried out. The comparison with the binary tree method proves that the algorithm is effective. In the 4th part, the compact difference scheme of three layers is constructed for nonlinear problems, and the prediction correction technique is used in the calculation of the first layer, and the numerical experiments are carried out and compared with the second order difference method. It is proved that the compact difference algorithm is more accurate than the second order difference algorithm.
【学位授予单位】：中国海洋大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F224;F830.9

【相似文献】