分形市场中分数阶导数期权定价模型的建立、解法与应用研究

项目名称： 分形市场中分数阶导数期权定价模型的建立、解法与应用研究

项目编号： No.71501031

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 管理科学

项目作者： 宋丽娜

作者单位： 东北财经大学

项目金额： 17.4万元

中文摘要： 期权定价问题是金融工程研究的核心问题之一。本项目以带有分数阶导数的Black-Scholes方程为模型，利用分数微积分和分数阶偏微分方程的理论和方法研究定价问题。首先，构建基于分数阶Black-Scholes方程的欧式和美式期权定价模型并求解，采用渐近分析、数值模拟和应用研究方法，探讨分数阶导数模型的可行性和有效性；其次，分别考虑混合分数布朗运动、随机利率和交易成本的情境，推导相应的分数阶导数模型并求解；最后，将分数阶导数模型引入权证市场，建立分数阶导数权证定价模型，针对市场数据进行应用研究，运用比较分析说明分数阶导数模型的适用性和优势。本项目体现现代数学方法和现代金融理论相互促进和发展，其结果为金融衍生品定价理论的研究提供新的思路、工具和方法。

中文关键词： 期权定价模型；衍生品；欧式期权；分形市场；分数阶偏微分方程

英文摘要： Option pricing is one of the key problem for financial engineering research.The project takes a Black-Scholes equation with fractional derivative as a model. It applies the theories and methods of fractional calculus and fractional partial differential equation to study the pricing problem. Firstly, European and American option pricing models based on the fractional Black-Scholes equation are constructed and solved. Using asymptotic analysis, numerical simulation and application research, the feasibility and effectiveness of the fractional derivative model are studied. Secondly, fractional derivative model are derived and solved under the situations of mixed fractional Brown motion, stochastic interest rates and transaction costs. Finaly, fractional derivative model is introduced to warrants market. Warrants pricing model with fractional derivative is etablished. The application researches are made based on market data. Comparison analysis explains the adaptation and superiority of fractional derivative model. The project reflects that the modern mathematical methods and the modern finance theory promote and develop each other. The results present new ideas, tools and methods for the theoretical research of financial derivatives pricing.

英文关键词： Option pricing model;Derivatives;European option ;Fractal market;Fractional partial differential equation