美式期权定价的近似解析法

发布时间：2018-01-18 11:45

本文关键词：美式期权定价的近似解析法　出处：《华南理工大学》2012年硕士论文　论文类型：学位论文

【摘要】：美式期权定价问题，可归结为最优停时问题或变分不等问题，一般没有闭形式的定价公式。对于这类期权的定价，通常采用：MonteCarlo(蒙特卡洛)模拟法[12,13]和半解析法[4,5]；最近，文献[22]提出了一种用渐近展式表示的、易于处理的美式期权定价方法。半解析法计算精确并且计算速度比较快，但很难拓展到Black-Scholes模型之外；文献[22]中方法的优点是它可以直接拓展到带有随机波动率和随机利率的多因素模型中，但该文献只给出了Black-Scholes模型下的三阶展式，而且也没有进行三元素拓展。借鉴文献[22]的思想，本文主要做了以下三方面工作：一是推导出了美式期权定价的近似解析法的一般递推公式；二是在利率和波动率都为扩散过程时，讨论Black-Scholes模型下的美式期权定价问题，给出了近似解析法的三阶递推公式；三是提供了一些具体实验及实验的结果，通过这些实验，可以看到这个方法的收敛性和精确性。本文得到三元素模型下递推公式的主要思想是：用一条近似最优执行规则代替最优执行规则。本文共分为四章：第1章，简要介绍了期权的相关概念以及美式期权定价的研究现况。第2章，，对Black-Scholes模型下美式期权定价问题的近似解析法做了必要的准备工作，引入了标准化币值(Normalizedmoneyness)的定义，并且讨论了它的一些直观的主要特征。第3章，详细推导了美式期权定价的近似解析法的一般递推公式，通过数据实验验证了它的收敛性和精确性。第4章，把我们的方法推广到含有随机波动率和随机利率的多因素模型中，并且进行了具体计算，计算结果显示这个方法对于合理的模型参数来说是精确的。
[Abstract]:The American option pricing problem, which can be attributed to the optimal stopping time problem or variational inequality problem, generally does not have a closed form of pricing formula for the pricing of such options. Usually we use the Monte Carlo simulation method. [12 ~ 13] and semi-analytical method. [4 / 5]; most recent, literature. [In this paper, an American option pricing method, which is expressed by asymptotic expansion and is easy to deal with, is proposed. The semi-analytical method is accurate and fast. But it is difficult to extend beyond the Black-Scholes model. Literature. [22] the advantage of this method is that it can be extended directly to the multivariate model with random volatility and stochastic interest rate, but only the third-order expansion under Black-Scholes model is given in this paper. And there is no expansion of the three elements. [The main work of this paper is as follows: first, the general recursive formula of the approximate analytical method of American option pricing is derived; Secondly, when the interest rate and volatility are diffusion process, the pricing problem of American option under Black-Scholes model is discussed, and the third-order recursive formula of approximate analytic method is given. The third is to provide some specific experiments and experimental results, through these experiments. The convergence and accuracy of this method can be seen. In this paper, the main idea of the recursive formula under the three-element model is to replace the optimal execution rule with an approximate optimal execution rule. This paper is divided into four chapters: chapter 1, briefly introduces the concept of options and the current situation of American option pricing. Chapter 2. In this paper, the approximate analytical method of American option pricing under Black-Scholes model is proposed. This paper introduces the definition of Normalized moneyness, and discusses some main intuitive features of it. Chapter 3. The general recursive formula of approximate analytical method for American option pricing is derived in detail. The convergence and accuracy of the formula are verified by data experiments. Chapter 4. Our method is extended to the multi-factor model with random volatility and stochastic interest rate. The calculation results show that the method is accurate for reasonable model parameters.
【学位授予单位】：华南理工大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F224;F830.9

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