Interest Rate Modeling at the Zero Lower Bound: Applications of Diffusions with Sticky Boundaries

自2008年全球金融危机以来，美国、欧元区和日本的短期利率一直接近于零，原因是各国央行对金融危机和随后的经济衰退采取了货币政策应对措施。当短期利率处于零利率下限时，传统的利率数学模型失效。本课题基于黏性边界扩散过程的数学，发展并研究了一类新的具有零下界的利率数学模型。该项目的预期影响是在金融行业中对利率敏感的金融工具的定价和套期保值，管理利率风险，固定收益投资组合构建，以及在中央银行中帮助实施货币政策，以及在金融数学和工程方面培养博士生。具有粘性边界的扩散类非常适合对ZLB建模的挑战，因为它自然地提供了具有两种不同经济制度的模型-远离边界的过程和边界上的过程。该项目开发了分析和计算工具来处理具有粘性边界的扩散，包括专门为这类随机过程量身定制的计算方法，并将其应用于开发和实证测试利率模型。本项目的智力优势在于开发了一类基于粘性边界扩散的新型利率模型，并提供了求解具有粘性边界的随机微分方程和具有Wentzell边界条件的偏微分方程的相关分析和计算方法。研究生也包括在这个项目中。

Short-term interest rates in the U.S., the Euro zone, and Japan have been near zero since the global financial crisis of 2008, due to the monetary policy responses by the central banks to the financial crisis and the recession that followed. Conventional mathematical models of interest rates break down when the short term interest rate is at the zero lower bound (ZLB). This project develops and investigates a novel class of mathematical models of interest rates with the zero lower bound based on the mathematics of diffusion processes with sticky boundaries. The anticipated impact of the project is in applications in the financial industry to the pricing and hedging of interest-rate-sensitive financial instruments, to managing interest rate risk, to fixed income portfolio construction, and in central banking to aid in conducting monetary policy, as well as in training of doctoral students in financial mathematics and engineering. The class of diffusions with sticky boundaries is well suited to the challenge of modeling the ZLB, as it naturally supplies a model with two distinct economic regimes -- the process away from the boundary and the process on the boundary. This project develops analytical and computational tools to work with diffusions with sticky boundaries, including computational methods specifically tailored for this class of stochastic processes, and applies them to develop and empirically test interest rate models. The intellectual merit of this project is in the development of a novel class of interest rate models based on diffusions with sticky boundaries, and in the associated analytical and computational methods to solve stochastic differential equations with sticky boundaries and partial differential equations with Wentzell boundary conditions. Graduate students are included in the project.