【作者】 唐耀宗；

【导师】 金朝嵩；

【作者基本信息】 重庆大学 ， 计算数学， 2008， 硕士

【摘要】 期权是最重要的金融衍生工具之一,它作为一种金融创新工具,在防范和规避风险以及投机中起着非常重要的作用。而如何通过合理的数学模型来确定期权的价格就成了投资者应用期权规避金融风险的关键性问题,所以在金融领域中,期权的定价问题成为理论和应用研究的一个重要领域。对于欧式期权,Black & Sholes早已给出解析形式的定价公式。然而,对于美式期权的定价,并不存在这样的解析公式,也无法求得精确解。而现实世界中,交易所中交易的大多数期权为美式期权。因此,发展各种计算美式期权价格的数值方法具有重要的理论和实际意义。美式期权定价问题的数学模型一般可归结为自由边值问题。本文绪论部分对金融衍生工具及其定价理论作了概括性的回顾;第二部分阐述了衍生证券价格所服从的Black—Scholes偏微分方程;第三部分对传统的二叉树、三叉树模型进行一些改进。即在适当的区域加密树图网格,而其他的区域不予变化,以消除粗糙树图网格中存在的“非线性误差”问题,而更能反映实际情况,使得只增加很少的计算量,就可以达到原本需要更高密度树图才能达到的计算效果。第四部分基于B-S微分方程,对有限差分方法中的一些参数设置方法进行改进,并将传统的内含有限差分法和外推有限差分法进行一定的结合,获得更好的结果。第五部分介绍了多项标的资产欧式期权所派生出来的标准形式,并使用基于对流扩散微分方程的基本解方法(MFS方法)求解派生出来的偏微分方程标准形式。在考虑了美式期权的特点与MFS方法的特性之后,将MFS方法推广到了美式期权的求解。第三、四、五每部分均采用了数值算例验证了该章方法的有效性和实用性。更多还原

【Abstract】 Option is one of the core tools of financial derivatives, which plays an important role in the effective management of risk and speculation. Risk management is depended on the right evaluation of option in a certain extent .The critical thing is how to value its fair price. A wide variety of the options traded in exchanges are American options. It is thus important to find appropriate ways to price American options. Thus, unlike European options, no explicit closed-form formulas have been found for American options, approximation methods have to be used in practice. Generally the classic Black-Scholes model for an American option leads to a free boundary problem with a degenerate partial differential operator.The part of this text introduction has done the reviewing of generality to the financial derivative and pricing theory. At the second chapter, the article expatiates the instauration of the Black-Scholes Differential Equation. At the third chapter of the article, nonlinearity error has been sharply reduced by grafting one or more small sections of fine high-resolution lattice onto the traditional Binomial Tree and Trinomial Tree with coarser time and price steps on the proper area. In the fourth chapter, the traditional differential scheme, based on the B-S differential equation, has been improved by changing some parameters presented, and collecting their both advantages. The following chapter presents the procedure for deriving canonical forms of European options of multi-assets is introduced. Then, the MFS based on the convection-diffusion fundamental solution is proposed to solve the resulted canonical forms. The MFS method has been expanded to the American option of single asset after considering the characteristics of MFS and American option. Numerical example were showed the efficiency and practicability of the algorithm at the end of the chapter 3, 4 & 5.更多还原