解美式期权和CEV模型下的美式期权的有限体积法

发布时间：2018-01-19 02:25

本文关键词： 美式期权 CEV模型 Front-Fixing方法有限体积法完全匹配层方法 Newton迭代　出处：《吉林大学》2014年硕士论文　论文类型：学位论文

【摘要】：本文主要介绍了一种解决美式期权定价问题的有限体积法,并进一步利用该方法解决不变方差弹性(CEV)模型下的美式期权定价问题。美式期权问题本质上是一个定义在无界的域上的自由边界问题,因此在应用数值方法解方程时会必然有截断的过程。本文中,我们在将原问题整理成线性形式后,先利用Front-Fixing方法将自由边界变为规则边界,再利用完全匹配层(PML)方法对无界边取截断。由于方程中含有未知的自由边界信息,所以在时间离散的每一层,我们采用Newton迭代,即在利用有限体积法解方程的同时通过Newton迭代得到最优的自由边界值,进而得到相应的期权价格。对于CEV模型下的美式期权问题,当弹性因子α1时,经过一定变换原问题可以转化成美式期权类似的形式,因此可以用相同的处理方法解决。对于α1的情况由于最终会转化成有界区域上的方程问题,相对简单,因此本文不做考虑。本文在给出以上两个问题的基本模型及相应算法之后,还将给出两个问题采用上述方法时问题的正定性证明,对于每个算法过程,其中的迭代步骤均可以总结为一个方程组求解问题,将其写为矩阵形式后,我们知道,如果右端为正,而左端系数矩阵满足M-矩阵定义,则所得的解就能保持正定性。通过以上所述步骤,我们可以最终保证我们的方法所得结果是正确并且合理的。在通过一系列对比数值实验的验证下,可以观察到该算法在实际应用中是有效的,而且适用于多种情况。该方法的数值结果与差分或有限元方法加细的结果吻合,并且在多数情况下,更光滑,就这一点来说较以往某些方法更接近事实,总体效果是令人满意的。随着时代的发展,各类期权的演化,关于美式期权及其变形问题的研究还将继续,对于其他形式的美式期权是否适用于本文的方法还不得而知,我们也将继续在该领域做出努力。
[Abstract]:This paper mainly introduces a finite volume method to solve the problem of American option pricing. Furthermore, this method is used to solve the American option pricing problem under the invariant variance elasticity (CEV) model. The American option problem is essentially a free boundary problem defined in an unbounded domain. Therefore, there must be a truncation process when solving the equation by numerical method. In this paper, we arrange the original problem into a linear form. Firstly, the free boundary is changed into a regular boundary by Front-Fixing method, and then the unbounded edge is truncated by the perfectly matched layer Front-Fixing method, because the equation contains unknown free boundary information. So in every layer of time discretization, we use Newton iteration, that is, we use the finite volume method to solve the equation and get the optimal free boundary value by Newton iteration. Then get the corresponding option price. For the American option problem under the CEV model, when the elastic factor 伪 is 1:00, the original problem can be transformed into a similar form of American option after a certain transformation. Therefore, we can solve the problem with the same method. For the case of 伪 1, it is relatively simple because it will eventually be transformed into the problem of equations in the bounded region. After giving the basic model and the corresponding algorithm of the above two problems, we will also give the positive qualitative proof of the two problems using the above method, for each algorithm process. The iterative steps can be summed up as a problem of solving equations, which is written as a matrix form, we know that if the right end is positive, and the left end coefficient matrix satisfies the definition of M- matrix. Then the obtained solution can keep the positive definiteness. Through the above steps, we can finally ensure that the results obtained by our method are correct and reasonable. It can be observed that the algorithm is effective in practical application and suitable for many cases. The numerical results of the method are in agreement with the finite-element method and are more smooth in most cases. This is closer to reality than some previous methods, and the overall effect is satisfactory. With the development of the times and the evolution of various kinds of options, the study on American options and their deformation will continue. It is not known whether other forms of American options are applicable to this paper. We will also continue to make efforts in this area.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：F224;F830.9

【共引文献】