基于不确定理论的亚式障碍期权定价研究

Research on Pricing Asian-Barrier Options Based on Uncertainty Theory

摘要: 亚式障碍期权是一种路径依赖型期权，其收益取决于股票价格在特定的到期时间能否到达预设的障碍水平。不同于传统的随机理论，本文将在不确定理论的框架内研究亚式障碍期权的定价问题。首先，通过假设标的股票价格服从不确定指数O-U过程，且利率服从不确定CIR过程，我们构建了一种新的不确定股票模型。然后，运用不确定微积分和不确定微分方程的相关知识，推导出新模型下亚式障碍期权的定价公式及相应的数值解。最后，运用矩估计法对不确定股票模型中的未知参数进行估值，并通过数值案例对得到的参数估计值进行了合理性验证。研究结果表明，带有浮动利率的不确定指数O-U股价模型更符合金融市场的实际情况，能够合理地对亚式障碍期权定价。

Abstract: Asian-barrier option is a path-dependent option, whose payment depends on whether the stock price can reach the pre-set barrier level at a specific maturity time. This paper will investigate the pricing issue of Asian-barrier options within the framework of uncertainty theory, which is distinct from traditional stochastic theory. Firstly, assuming that the underlying stock prices follow an uncertain exponential O-U process and the interest rates follow an uncertain CIR process, we have constructed a new uncertain stock model. Then, the pricing formula and the corresponding numerical solution of Asian-barrier options are derived, based on the knowledge of uncertain calculus and uncertain differential equation. Finally, we apply the method of moments to estimate the unknown parameters in the uncertain stock model and validate the reasonableness of the estimated parameters through numerical examples. The research results indicate that the uncertainty exponential O-U stock price model with floating interest rates is more in line with the actual situation of the financial market and can reasonably price Asian barrier options.