【作者】 董志华；

【导师】 王建华；

【作者基本信息】 武汉理工大学 ， 数量经济学， 2010， 硕士

【摘要】 Fischer Black和Myron Scholes在1973年推导出了基于无红利支付股票期权的Black-Scholes期权定价方程(B-S方程),并精确地给出了欧式期权定价的解析表达式,但对于美式期权,由于存在提前执行的可能性,所以只能找到方程近似解析解或是数值解。有限元差分方法是美式期权定价中较为常见的数值计算方法,它对离散后的方程在有限的区域进行差分。由于差分区域的不规则性,在差分区域的边界往往存在相对较大的截断误差。本文中利用美式期权和欧式期权的关系,通过适当的变量代换,简化含自由边界的抛物线偏微分方程的美式期权定价模型,使得在数值计算中进行有限差分的区域变为规则形状,消除初始条件的奇点,进而减少数值计算的截断误差。B-S方程是期权定价的基础。对于标的物支付连续红利的欧式期权,在假设标的物价格服从几何布朗运动的前提下,我们详细推导了B-S方程。基于B-S方程,我们分析了美式期权的定价问题,其本质是一个障碍问题。以美式看跌期权为例,对于任意一个到期日之前的时间,存在一个相应的标的物价格,在此价格的左侧,美式看跌期权的价值始终会等于相应的价值函数,而当标的物价格高于此价格时,期权价格会大于相应的价值函数。这个临界的标的物价格是依赖于时间的,这就导致了一个自由边界,从而使得难以找到方程的精确的解析解,寻找美式期权数值解便成为研究其解特性的主要方法。我们详细介绍了奇点分离法的主要思想,分析了它在数值计算方法中的应用。由于欧式期权可以准确得到,利用美式和欧式期权的关系,我们考虑两者之间的差而不是直接求解美式期权的价格,这样能减少相对误差。由于美式期权定价方程中的边界条不够光滑,我们通过引入适当的变量代换,使得方程的边界条件足够的光滑,进而减少数值计算的截断误差。二叉树法及投影SOR法是两种常用的美式期权定价的数值方法,通过数值实验与并这的两种数值方法进行比较,发现奇点分离法能明显改善计算精度和速度。更多还原

【Abstract】 In 1973, based on non-dividend-paying stocks option, Fischer Black and Myron Scholes derived Black-Scholes option pricing formula and gave a precise analytical solution for European option. However, due to the possibility of early exercise of American option, we could only find the approximated analytical solution or, mostly the numerical solution in stead of analytical one. Finite difference method which solves discrete equation on a finite area, is one of the popular methods in American option pricing. The irregularity of the difference area may lead to relatively big truncation error. By making use of the relation between the American option and the corresponding European option and introducing some appropriate variable transformations, in this paper, we simplify the pricing model for American option which involves a parabolic equation with free boundary condition. These simplifications convert the original area where to take difference into a regular shape and eliminate the singularity in the initial condition. Moreover, the separation of singularity decreases the truncation error of numerical computation.B-S equation is the pillar of the option pricing. In this paper, assuming that the price of the stock follows the geometric Brownian motion, we deduce the B-S equation in details. Based on this equation, we study the pricing model for American option which essentially is an obstacle problem. Take the American put option as an example, for any particular time before expiration, there exists a corresponding price for which the value of the American put option equals the corresponding payoff function if the real price is less than the fixed price otherwise the value of the put option is greater than the payoff function. This particular price depends on time, which leads to a so called free boundary. Due to the uncertainty of this boundary, it is hard to find the analytical solution of the American option. Therefore, numerical methods become the main approaches to study the pricing of American option. The fundamental idea of singularity separation method is discussed in details. Because the Europe option price could be obtained precisely, we consider the difference of the price of American and the corresponding Europe option instead of the price of the American option itself, which could decrease the relative error. Moreover, the bound of the equation of the American option pricing is not smooth enough. Therefore, we introduce some appropriate variable transformation in order to control the truncation error. The binomial method and Projected SOR method are two normal method in American option pricing, we compare the singularity separation method with the these two and find that singularity separation method efficiently improves the accuracy and the speed of computation.更多还原