【作者】 吴志刚；

【导师】 金朝嵩；

【作者基本信息】 重庆大学 ， 计算数学， 2001， 硕士

【摘要】 “金融数学、金融工程和金融管理”是国家自然科学基金委员会确定的我国九五期间的重大研究项目，而期权定价理论则是目前金融工程、金融数学所研究的前沿和热点问题。文章在深入研究了期权特性及其价格影响因素的基础上，运用相关的数学工具，推导出了一种新型的期权定价模型，并给出了其数值计算方法。文章首先从著名的布莱克—斯科尔斯期权定价公式(B—S定价公式)入手，详细地阐述了衍生证券价格所服从的B—S偏微分方程的推导过程，以及由B—S方程推出B—S定价公式的方法，并分析了这个定价公式的不足之处。针对这个不足，文章将在改变B—S定价公式中一个基本假设的基础上，推导出一种新型的期权定价模型。首先，在第三章以股票为例详细分析了衍生产品标的物价格的走势状况，先分别对马尔科夫跳跃过程和Ito过程进行了较为详细的分析，而后在经典的股票价格服从对数正态分布的假设基础之上，加入了马尔科夫跳跃过程，将二者复合后得到了一个新型的混合过程，并推导出了一个股价走势的分布模型，即假设股票价格的行为模式是一个混合过程。在第三章的后半部分，文章对深沪两地股市的综合股价指数以及深市中一支股票(深能源)的实际价格进行计算，以实际数据验证了混合分布相对于对数正态分布的优势性。基于这个分布模型，在第四章文章进而推出了标的股票价格服从混合分布的衍生证券价格所服从的偏微分方程。然后，沿用在第三章中推出混合分布模型的思路，又导出了欧式看涨期权的定价公式，来作为文章的主要结果。注意到马尔科夫跳跃过程的间断性，该定价公式也是分时段型的。在文章的最后部分，研究了数值方法在期权定价方面的应用，运用数值方法求解了前文所推出的衍生证券价格所服从的偏微分方程，并得到了可行的算法，其中主要用到的方法有有限差分法和有限元法。更多还原

【Abstract】 "Financial Mathematics, Financial Engineering and Financial Managing" is one of the important items during the ninth Five-Year-Plan in China, and option pricing is being studied heartily in this item. Base on the detailed research of the characters of option and the affecting factors of option’s price, the article gives us a new option pricing model and its numerical method using the mathematical tools. Firstly, the article studies the famous Black-Scholes Option Pricing Model in detail, including the inferring process of the B-S Differential Equation, and the method how to infer the B-S Model from the B-S Differential Equation. Secondly, the article gives us one of the important faults of this model, hi view of this fault, the article will infer a new type of option pricing model by means of changing one of the basic hypothesizes of B-S Model. In the third chapter, the article studies stock’s changing tendency, researching the Markov jump process and Ito process. Afterwards, the article refers a new mixed process by mixing the two processes, saying that the stock’s changing tendency is following the mixed process. At the last section of the third chapter, the article analyze the SHENZHEN and SHANGHAI stock markets’ synthetic number and a piece of stock’s (Shen-Nengyuan) number, proving the superiority of the mixed process. Base on this distribution model, the article infers the differential equation of derivative security whose basic stock’s distribution is mixed process, the pricing model of European Call Option. Noting the Markov jump process is not continuous; this pricing model is also interrupted in view of time. At the end of this article, we solve the differential equation with the finite difference method and the finite element method, obtaining an executable method.更多还原