欧式向上敲出指数障碍看跌期权的定价

发布时间：2018-11-19 09:04

【摘要】：著名的Black-Scholes股票期权定价模型是期权定价理论的重要内容之一,是金融工程研究的基础.虽然得到了股票期权定价模型所满足的偏微分方程(Black-Scholes-Merton微分方程),求出了一些期权的解析定价公式.但是在许多情况下,并不能求出某些奇异期权的定价公式,这就需要用数值方法计算期权价值.期权定价的数值方法有多种形式,研究他们的收敛速度对期权定价是非常重要的.本文主要研究欧式向上敲出指数障碍期权定价的解析方法和三种数值方法,并比较不同数值方法得到的数值解收敛于解析解的速度. 第一部分,主要运用Green Function的方法得到了本文所研究的期权的解析解: 第二部分,探讨了三种数值方法:显式有限差分法,隐式有限差分法, Crank-Nicolson法.这三种方法都需要把连续问题离散化,离散方式不同,得到的数值方法不同.首先,对定义域离散化,在定义域的区域做网格划分,用有限个网格节点代替连续区域.其次,对偏微分方程离散化,用差分因子代替微分因子,将微分方程变成差分方程.利用边界条件,通过迭代来求出差分方程的解,从而得到期权的数值解.通过不断加密定义域网格,可得到不同的数值解.通过具体数据,对这三种数值方法的收敛性做了比较,当障碍水平大于敲定价格时,对本文所研究的期权来说,显式有限差分法的收敛速度最快. 第三部分,讨论了障碍水平与敲定价格之间的大小关系对数值解收敛速度的影响,通过具体数据分析了每一种情况下的数值解的收敛速度.当障碍水平大于敲定价格时,显式有限差分法收敛于解析解的速度较快;当障碍水平与敲定价格有一个交点时,三种数值解收敛于解析解的速度减慢;当障碍水平小于敲定价格时,三种数值方法得到的数值解总体上是收敛的趋势,收敛速度差不多.
[Abstract]:The famous Black-Scholes stock option pricing model is one of the important contents of option pricing theory and the foundation of financial engineering research. Although the partial differential equation (Black-Scholes-Merton differential equation) satisfied by the stock option pricing model is obtained, some analytical pricing formulas of the stock option are obtained. However, in many cases, it is impossible to find out the pricing formula of some strange options, which requires the numerical method to calculate the value of the options. There are many kinds of numerical methods for option pricing, and it is very important to study their convergence rate. In this paper, we mainly study the analytical methods and three numerical methods for pricing the Euclidean upward knock out exponential barrier options, and compare the speed of convergence of the numerical solutions obtained by different numerical methods to the analytical solutions. In the first part, we mainly use Green Function's method to get the analytical solution of options studied in this paper. In the second part, we discuss three numerical methods: explicit finite difference method, implicit finite difference method and Crank-Nicolson method. These three methods need to discretize the continuous problems. The numerical methods are different in different ways. Firstly, the domain is discretized and meshed, and a finite number of mesh nodes are used to replace the continuous region. Secondly, for the discretization of partial differential equation, the difference factor is used to replace the differential factor, and the differential equation is transformed into the difference equation. By using the boundary condition, the solution of the difference equation is obtained by iteration, and the numerical solution of the option is obtained. Different numerical solutions can be obtained by continuously encrypting domain mesh. The convergence of these three numerical methods is compared by specific data. When the barrier level is greater than the fixed price, the explicit finite-difference method has the fastest convergence rate for the options studied in this paper. In the third part, the influence of the relationship between the barrier level and the fixed price on the convergence rate of the numerical solution is discussed, and the convergence rate of the numerical solution in each case is analyzed by the specific data. When the barrier level is greater than the fixed price, the explicit finite-difference method converges to the analytical solution faster, and when the barrier level has an intersection point with the firm price, the speed of the three numerical solutions converges to the analytical solution. When the barrier level is less than the fixed price, the numerical solutions obtained by the three numerical methods are convergent, and the convergence rate is similar.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F224;F830.9

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