美式期权定价的一种快速算法

发布时间：2018-03-30 08:18

本文选题：美式期权定价　切入点：数值方法　出处：《上海交通大学》2012年硕士论文

【摘要】：期权是人们为了规避市场风险而设计出来的一种金融衍生工具。期权定价是金融衍生工具理论研究和实际应用的核心。期权定价理论是目前金融工程、金融数学研究的前沿和热点问题。期权作为一种衍生产品，它的定价和模型取决于原生资产价格的演化模型。在连续时间情形原生资产价格演化可以通过一个随机微分方程来描述，从而在此基础上，作为它的衍生物-期权的价格适合的是一个偏微分方程的定解问题。因此把偏微分方程作为工具，利用偏微分方程的理论和方法、建立各种期权定价的数学模型，导出期权的定价公式，对期权的价格结构作深入的定性分析以及利用偏微分方程数值分析方法给出求期权价格的算法是可行的。美式期权可以提前执行，在实践中具有更大的灵活性。一般情况下，美式期权价格却没有精确的解析定价公式，因此研究美式期权定价问题的数值解法具有重要的意义。本文研究了一种较快速的数值解法。主要内容如下：第一章简要介绍了期权知识、美式期权定价问题模型与美式期权价格的性质。第二章介绍一种无界区域上的热传导方程的数值方法。我们先引进一个人工边界条件Γ,然后提出一个精确的人工边界条件使得原问题成为有限区域上的热传导方程。于是有限差分法可以用来解决有限计算区域的问题。第三章介绍线方法。第四章通过人工边界法解决半无限区域的一维Black-Scholes的边界条件，再用线方法寻找自由边界从而比较快速地解决一维美式期权的定价问题
[Abstract]:Option is a kind of financial derivative designed by people to avoid market risk. Option pricing is the core of the theoretical research and practical application of financial derivatives. Option pricing theory is the current financial engineering. The frontier and hot issues of financial mathematics research. Options as a derivative product, Its pricing and model depend on the evolution model of primordial asset price, which can be described by a stochastic differential equation in continuous time. The price of options, as its derivative, is suitable for the definite solution of a partial differential equation. Therefore, taking the partial differential equation as a tool and using the theory and method of partial differential equation, the mathematical models of option pricing are established. It is feasible to derive the pricing formula of option, to make a deep qualitative analysis on the price structure of option and to give the algorithm of calculating option price by using partial differential equation numerical analysis method. In practice, there is more flexibility. In general, there is no exact analytical pricing formula for American option price, so it is of great significance to study the numerical solution of American option pricing problem. In this paper, a fast numerical method is studied. The main contents are as follows:. The first chapter briefly introduces the option knowledge, the model of American option pricing problem and the nature of American option price. In the second chapter, we introduce a numerical method of heat conduction equation in unbounded region. We first introduce an artificial boundary condition 螕, then we propose an exact artificial boundary condition to make the original problem become heat conduction in a finite region. The finite difference method can be used to solve the problem of finite computational domain. The third chapter introduces the line method. In chapter 4, the boundary conditions of one-dimensional Black-Scholes in semi-infinite region are solved by artificial boundary method, and then the free boundary is found by line method to solve the pricing problem of one-dimensional American option more quickly.
【学位授予单位】：上海交通大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F224;F830.9

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