期权定价中的重点抽样蒙特卡洛模拟

发布时间：2018-03-20 23:32

本文选题：蒙特卡洛模拟　切入点：方差缩减　出处：《苏州大学》2012年硕士论文　论文类型：学位论文

【摘要】：在这篇论文里,我们研究了用于缩减金融衍生品定价过程中方差的重点抽样蒙特卡洛方法,这些衍生品都由高维高斯向量驱动。正如P.Glasserman论文中的重要结果所说明的那样,首先要采用测度变换,在测度变换中得到决定最优漂移系数的方程组,然后使得模拟中对应于新测度的方差达到最小。本文给出了最优漂移系数的存在性和唯一性的证明,并应用了有效的牛顿算法寻找这个漂移系数。我们通过几个期权定价问题说明了我们方法的有效性,这些期权包括：标的资产服从几何布朗运动的亚式期权,Heston模型,跨式期权和一篮子期权。另外,在此适用的分层抽样技术也被用来得到更大程度的方差缩减。这个方法的一个优点就是我们无需对收益函数的光滑性做任何要求,即便我们不知道收益函数的准确表达式；另外一个优点就是最优漂移系数是唯一的,这使得模拟中的方差真正地达到最小。值得一提的是,在多峰问题中,漂移重点抽样的方差缩减效果并不是很好。我们需要改进用于重点抽样的密度函数,比如我们可以使用混合正态密度,还可以使用t分布密度,也可以考虑对正态密度进行尺度变换(引入第二个参数σ,即标准差)等等这些都是我们今后工作的方向。当然,如果原生资产服从Levy过程,我们同样也可以考虑Levy过程下的漂移变换重点抽样,从而缩减蒙特卡洛模拟的方差。本文安排如下：本文第一章简要介绍Monte Carlo方法的背景知识。第二章介绍如何通过模拟产生样本路径,从而模拟期权的价值和方差。在第三章我们建立本文的模型并讨论了方法的可行性。在第四章我们对所建立的模型以及已有的有关模型进行实证分析。在最后一章,我们对本文进行总结并给出后继研究建议。
[Abstract]:In this paper, we study the Monte Carlo method for reducing variance in the pricing of financial derivatives, which are all driven by high-dimensional Gao Si vector, as the important results in P. Glasserman's paper illustrate. First, the measure transformation is used to obtain the equations that determine the optimal drift coefficient in the measure transformation, and then the variance corresponding to the new measure in the simulation is minimized. In this paper, the existence and uniqueness of the optimal drift coefficient are proved. The effective Newton algorithm is used to find the drift coefficient. Several options pricing problems are used to illustrate the effectiveness of our method. These options include: the Heston model of the Asian option of the underlying asset from the geometric Brownian motion. Cross options and a basket of options. In addition, the stratified sampling technique applicable here is also used to achieve a greater reduction in variance. One advantage of this approach is that we do not have to make any demands on the smoothness of the return function. Even if we do not know the exact expression of the return function another advantage is that the optimal drift coefficient is unique which makes the variance in the simulation really minimum. It is worth mentioning that the variance reduction effect of drift focus sampling is not very good in multi-peak problems. We need to improve the density function for key sampling, for example, we can use mixed normal density. We can also use t distribution density, we can also consider the scale transformation of normal density (introducing the second parameter 蟽, that is, standard deviation) and so on, which are the direction of our future work. Of course, if the original assets are taken from the Levy process, We can also reduce the variance of Monte Carlo simulation by taking into account the shift sampling in the Levy process. This paper is organized as follows: the first chapter briefly introduces the background of Monte Carlo method, the second chapter introduces how to generate sample path by simulation. In Chapter 3, we establish the model and discuss the feasibility of the method. In Chapter 4th, we make an empirical analysis of the established model and the existing models. In the last chapter, We summarize this paper and give some suggestions for follow-up research.
【学位授予单位】：苏州大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F830.9;F224

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