基于特征函数与数值计算的亚式期权定价

发布时间：2018-01-28 07:19

本文关键词： 特征函数等鞅变换数值计算　出处：《厦门大学》2014年硕士论文　论文类型：学位论文

【摘要】：亚式期权非常流行,它是最受欢迎的新型期权之一。虽然亚式期权经过了众多的学者与研究者反复的研究,但是至今也没有形成一个统一的定价表达式。在研究的过程中,研究者们也提出了许多的定价方法,现在为大家所接受的定价方法有几何平均近似算术平均法、蒙特卡洛模拟法、偏微分方程数值解法、特征函数法。作为学术界研究的重要课题的亚式期权是路径依赖的期权,因此其定价问题变得异常的复杂,这也使得它成为了研究中的难题。本文在特征函数法和数值计算的基础上,推导了亚式期权价格的表达式,得到了数值结果,为亚式期权的定价提供了新的思路。本文的创新之处在于在标的资产价格变化中引入了跳跃与波动率的变化,并且在用特征函数进行定价的基础上,利用数值计算的方法讨论了期权价与跳跃、波动率的关系。跳跃是无处不在的,然而跳跃又具有随机性与偶然性,这使得它成为生活中极其重要的现象,也使得它成为了学术研究中极为重要的研究课题。波动率随着时间的变化而变化的观点得到了学术界的认可,同时也为实际生活中的数据所证实。本文采用等鞅变换得到等价的随机过程,然后利用特征函数法得到两组常微分方程组和亚式期权价格的形式上的表达式,最后通过利用数值计算的方法得到了期权价格。在特征函数法与等鞅变换的基础上,我们得到了亚式期权价格的表达式。我们也通过数值计算的方法来研究了期权价格与各种变量的变化关系,在设定一系列的初值后,本文通过微分方程的数值解法得到了常微分方程组的数值解,同时通过数值积分得到了亚式期权的价格。在数值计算的基础上,我们得到以下结论：第一,跳跃的存在会影响亚式期权的价格;第二,跳跃强度的大小会影响亚式期权的价格；第三,波动率的变化会影响期权价格。
[Abstract]:Asian option is very popular, it is one of the most popular new options, although Asian option has been studied repeatedly by many scholars and researchers. However, there is not a unified pricing expression. In the process of research, researchers also put forward many pricing methods, which are now accepted as geometric mean approximate arithmetic average. Monte Carlo simulation method, partial differential equation numerical solution, characteristic function method. As an important subject of academic research, Asian option is path dependent option, so the pricing problem becomes very complicated. Based on the eigenfunction method and numerical calculation, the expression of the price of Asian option is derived, and the numerical results are obtained. It provides a new idea for the pricing of Asian options. The innovation of this paper lies in the introduction of jump and volatility in the change of underlying asset price, and on the basis of pricing with characteristic function, the paper discusses the option price and jump by numerical calculation method. The relationship of volatility. Jump is everywhere, but jump has randomness and contingency, which makes it an extremely important phenomenon in life. It has also become an extremely important research topic in academic research. The point of view that volatility changes with time has been recognized by the academic community. In this paper, the equivalent stochastic process is obtained by using the equal-martingale transformation, and then the formal expressions of two sets of ordinary differential equations and the price of the Asian option are obtained by using the eigenfunction method. Finally, the option price is obtained by numerical calculation. On the basis of the eigenfunction method and the equal-martingale transformation, we obtain the expression of the price of the Asian option. We also study the relationship between the price of the option and the variables by numerical calculation. After a series of initial values are set, the numerical solution of ordinary differential equation system is obtained by numerical solution of differential equation, and the price of Asian option is obtained by numerical integration. We get the following conclusions: first, the existence of jump will affect the price of Asian options; Secondly, the price of Asian option will be affected by the strength of jump. Third, volatility changes will affect option prices.
【学位授予单位】：厦门大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：F224;F830.91

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