The original Ait-Sahalia model of the spot interest rate proposed by Ait-Sahalia assumes constant volatility. As supported by several empirical studies, volatility is never constant in most financial markets. From application viewpoint, it is important we generalise the Ait-Sahalia model to incorporate volatility as a function of delay in the spot rate. In this paper, we study analytical properties for the true solution of this model and construct several new techniques of the truncated Euler–Maruyama (EM) method to study properties of the numerical solutions under the local Lipschitz condition plus Khasminskii-type condition. Finally, we justify that the truncated EM approximate solution can be used within a Monte Carlo scheme for numerical valuations of some financial instruments such as options and bonds.

Ait-Sahalia提出的原始Ait-Sahalia即期利率模型假定波动率恒定。正如一些经验研究所支持的那样，在大多数金融市场中，波动率从来都不是恒定不变的。从应用的角度来看，重要的是我们推广Ait-Sahalia模型以将波动率作为现货汇率延迟的函数。在本文中，我们研究了该模型的真实解的解析性质，并构建了几种截断的Euler–Maruyama（EM）方法的新技术来研究局部Lipschitz条件和Khasminskii类型条件下数值解的性质。最后，我们证明可以在蒙特卡洛方案中使用截短的EM近似解来对某些金融工具（例如，期权和债券）进行数字估值。