数学金融的分数次Black-Scholes模型及应用

发布时间：2018-05-19 02:31

本文选题：分数次布朗运动 + 衍生证券　；参考：《湖南师范大学》2004年博士论文

【摘要】：1973年,两位伟大的金融理论家与实务家Fisher Black和MyronScholes发表了他们的著名论文“期权定价与公司债务”(The pricingof options and corporate liability),给出了欧式期权定价的显式表达式,即著名的Black-Scholes公式。这是现代金融数学的一项具有里程碑意义的突破性成果。从此,金融数学的研究得到了蓬勃的发展,取得了非常丰硕的成果。特别是Black-Scholes模型,不仅在理论研究上出现了一大批成果,而且应用于金融市场,受到广泛的欢迎。20世纪90年代,全世界金融衍生证券市场每年的交易量已达50万亿美元。本文在分数次布朗运动的积分理论的基础上,对数学金融的具有任意Hurst参数的分数次Black-Scholes模型进行了全面系统的研究。在绪论中,介绍了金融数学的发展历史,特别对期权定价理论主要研究内容、成果及目前研究热点进行了较为详尽的综述。第二章,我们首先介绍了分数次布朗运动的定义、性质及其积分理论的主要成果。然后,给出了拟-条件期望及拟-鞅的定义,并得到了分数次布朗运动函数的拟-条件期望的计算公式。在第三章中,我们研究了分数次Black-Scholes模型,提出了利用拟-条件期望的分数次风险中性定价;得到了不同条件下欧式未定权益在到期前任意时刻的一般定价公式;并在无风险利率和红利率为时间的非随机函数的条件下,求出了欧式期权在到期前任意时刻的定价公式。第四章,我们研究了几种常见的奇异期权,在无风险利率和红利率分别为常数或为时间的非随机函数的条件下,得到欧式双向期权、混合期权、上限型买权、抵付型买权、后定选择权在到期前任意时刻的定价公式。在第五章中,我们研究了多维分数次Black-Scholes模型,在单资产多噪声、多资产单噪声和多资产多噪声三种情况下,得到欧式未定权益在初始时刻的定价公式。第六章,我们讨论了分数次Black-Scholes模型中的最优资产组合和最优消费资产组合问题,得到在给定效应函数条件下的最优资产组合和最优消费资产组合问题的显式解。
[Abstract]:In 1973, two great financial theorists and practitioners, Fisher Black and MyronScholes, published their famous paper "option pricing and Corporate debt" and gave an explicit expression of European option pricing, known as the Black-Scholes formula. This is a landmark breakthrough in modern financial mathematics. Since then, the financial mathematics research has obtained the vigorous development, has obtained the very rich achievement. Especially the Black-Scholes model, which not only has a large number of achievements in theoretical research, but also has been widely used in the financial market. In the 1990s, the transaction volume of the financial derivative securities market in the world has reached 50 trillion US dollars every year. Based on the integral theory of fractional Brownian motion, the fractional Black-Scholes model with arbitrary Hurst parameters is studied in this paper. In the introduction, the development history of financial mathematics is introduced, especially the main research contents, achievements and current research hotspots of option pricing theory are summarized in detail. In chapter 2, we first introduce the definition, properties and main results of integral theory of fractional Brownian motion. Then, the definitions of quasi-conditional expectation and quasi-martingale are given, and the formula of quasi-conditional expectation of fractional Brownian motion function is obtained. In chapter 3, we study fractional Black-Scholes model, propose fractional risk neutral pricing using quasi-conditional expectation, obtain the general pricing formula of European contingent claims at any time before expiration under different conditions. Under the condition that risk-free interest rate and red interest rate are non-random functions of time, the pricing formula of European option at any time before maturity is obtained. In chapter 4, we study several common singular options. Under the condition that risk-free interest rate and red interest rate are constant or non-random functions of time, we obtain European two-way options, mixed options, upper limit options, offset buying rights. The pricing formula for a postfixed option at any time before expiration. In chapter 5, we study the multi-dimensional fractional Black-Scholes model. In the case of multi-asset multi-noise, multi-asset single-noise and multi-asset multi-noise, we obtain the pricing formula of European contingent equity at the initial moment. In chapter 6, we discuss the optimal portfolio problem and the optimal consumption portfolio problem in fractional Black-Scholes model. We obtain the explicit solution of the optimal portfolio and optimal consumer portfolio problem under the condition of given effect function.
【学位授予单位】：湖南师范大学
【学位级别】：博士
【学位授予年份】：2004
【分类号】：F224

【引证文献】