美式期权的定价方法介绍与比较

发布时间：2018-03-01 05:14

本文关键词： 美式期权 Black-scholes模型自由边界有限差分法二叉树方法 LSM　出处：《山东大学》2012年硕士论文　论文类型：学位论文

【摘要】：本文第一章是绪论部分,简要介绍了期权的定义与分类情况以及期权定价的理论基础,其中主要介绍了风险中性原理、无套利定价原理和期权的价值分析,在这一章的最后对期权定价理论从无到有,从简到繁的发展过程做了一番介绍,一代一代的专家学者都为此作出了卓越的贡献。由于Black与Scholes作出的尤其突出的贡献,我们在第二章介绍了Black-Scholes模型的前提假设、建立过程以及定价公式的推导过程,重温了Black与Scholes的工作,另一方面,我们也发现了Black-Scholes模型的缺点：只适用于欧式期权,但是我们本文研究的是美式期权,即便如此,我们的工作并不是一无是处,在接下来的第三章我们重点介绍的是美式期权的定价模型,首先明确说明美式期权的定价问题属于自由边界问题,根据美式期权的特点我们建立了两种形式的定价模型：抛物型方程模型和变分不等式方程模型。在本章的最后一节我们又推出了美式期权定价的看涨—看跌对称关系。光有模型不行,还需要求解方法,为此,在接下来的第四、五、六章分别介绍了美式期权的三种解法：有限差分法、二叉树方法、最小二乘蒙特卡洛模拟方法。有限差分法是先介绍了差分方法,然后基于变分不等式模型的离散化进行了推导,得出了显示差分格式和隐式差分格式,其中显示差分格式还给出了Matlab程序,以及一个数值算例,最后给出了差分方法的评价。二叉树方法那一章先介绍了二叉树方法的定义,然后给出了二叉树的定价过程,再根据美式期权的特点给出了美式期权的定价过程,最后给出了方法评价,这一章属于介绍性质的内容,原因是二叉树方法中参数的选择不确定,在以往的研究中有几位优秀的专家学者分别给出了不一样的参数选取方法,只能具体情况具体分析了,再加上二叉树方法的“维数效应”问题,所以应用很有限,本文只做这些介绍而已。最小二乘蒙特卡洛方法是最近出现的新方法,不是我的原创,但是很新颖,应用起来也很方便,很广泛就把它也加入进来了,详细地介绍之后,自己进行了Matlab编程,也进行了数值实验,最后给出了方法评价。做了这么多工作之后发现,本篇论文的结构是：期权介绍,B-S模型,美式期权模型的建立,美式期权定价的求解方法。
[Abstract]:The first chapter is the introduction, which briefly introduces the definition and classification of options and the theoretical basis of option pricing, including risk neutral principle, no-arbitrage pricing principle and option value analysis. At the end of this chapter, the theory of option pricing has been introduced from scratch, from simplicity to complexity, and a generation of experts and scholars have made outstanding contributions to it. Because of the outstanding contribution made by Black and Scholes, in the second chapter, we introduce the premise hypothesis of Black-Scholes model, the establishment process and the derivation process of pricing formula, review the work of Black and Scholes, on the other hand, We also found the disadvantage of the Black-Scholes model: only for European options, but we are studying American options in this paper, but even so, our work is not useless. In the following chapter, we focus on the pricing model of American option. Firstly, we clearly explain that the pricing problem of American option belongs to the free boundary problem. According to the characteristics of American option, we establish two kinds of pricing models: parabolic equation model and variational inequality equation model. In the last section of this chapter, we derive the bullish-bear-bear-symmetry relation of American option pricing. For this reason, in the following chapters 4th, 5 and 6, three solutions of American option are introduced: finite difference method, binary tree method and least square Monte Carlo simulation method. The finite difference method is introduced first, then based on the discretization of variational inequality model, the display difference scheme and implicit difference scheme are derived. The display difference scheme and the Matlab program are also given. And a numerical example is given. Finally, the evaluation of the difference method is given. In the chapter of binary tree method, the definition of binary tree method is introduced, then the pricing process of binary tree is given, and then the pricing process of American option is given according to the characteristics of American option. Finally, the method evaluation is given. This chapter belongs to the content of introduction, the reason is that the choice of parameters in binary tree method is uncertain. In the past research, several excellent experts and scholars have given different parameter selection methods, which can only be analyzed concretely. In addition, due to the "dimension effect" problem of binary tree method, the application is very limited. The least square Monte Carlo method is a new method that has recently emerged, not my original one, but it is very novel and convenient to use. It is also widely included. After a detailed introduction, I have carried out Matlab programming myself. Numerical experiments are also carried out, and the evaluation method is given. After doing so much work, it is found that the structure of this paper is: option introduction to B-S model, the establishment of American option model, the solution of American option pricing.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F224;F830.9

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