基于罚方法的美式期权定价问题的数值计算

In the financial market, the option is an important tool of risk management. As the financial derivatives pricing based on Black-Scholes option pricing formula on the development of financial derivatives has played an important role. However, under the framework of Black-Scholes model, the European option price analytical solutions can be obtained，but American option is a free-boundary problem involving a partial differential equation，no closed-form solution. Therefore, centering the option pricing theory and method, American option pricing problem is investigated. The complementarity problem which the price meets basing on a partial differential operator is analyzed and discussed, and the nonlinear partial differential equations is obtained after the introduction of penalty method. Considering the convection situation, the local fitting technique is used to reasonable approximate the convection and second-order mixed derivatives, and the upwind technique is employed to avoid shock of the numerical solution, the numerical solution of partial differential equations are solved. Finally, the stability and convergence of the corresponding numerical schemes are discussed through a numerical example of the non standard American option-barrier option, American options under stochastic interest rates and so on.

在金融市场中，期权是风险管理的重要工具。作为金融衍生品定价的基础Black-Scholes期权定价公式对金融衍生品的发展起了不可估量的作用。然而，在Black-Scholes模型框架下，欧式期权价格可以获得解析解形式，而美式期权是一个偏微分方程自由边界问题，没有闭形式解。为此，以期权定价理论方法为中心，研究美式期权定价问题，分析和讨论其价格满足基于一个偏微分算子的互补问题，引入罚方法，使其转化为非线性偏微分方程。考虑到对流占优情形存在，采用局部拟合技术合理近似对流项和二阶混合导数项，应用迎风技巧避免数值解的震荡，求解该偏微分方程数值解。最后，通过数值算例研究非标准的美式期权-障碍期权、随机利率下美式期权等对应数值格式的稳定性和收敛性。

金融问题一直都是学术界研究的热点对象。由此衍生的一些数值方法受到众多学者的关注。研究衍生品定价时标的资产的动态演化过程是相当重要。从能源衍生品、天气衍生品等入手，运用随机理论和微分方程的知识，首先探讨了确定性系统和随机系统的稳定性和复杂性。基于随机分析理论，罚方法和拟合有限体积法耦合，把线性互补问题限制到有限区域，对时间、空间区域进行离散，给出算法并进行数值求解，通过数值实验，验证算法是高效性。本项目解决或部分解决了期权定价的随机理论、确定性演化过程的复杂性、分数布朗运动下期权定价等，寻求把抽象微分和随机理论用于建立期权定价理论的数学基础，促进了金融领域某些问题的解决，已发表或录用论文4篇。