When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing a third-order Robin boundary condition. Rather than computing the optimal exercise boundary from the analytical approximation, we simply obtain it from the asset option based on the linear relationship at the left boundary. As such, a high order convergence rate can be achieved. We validate by examples that the improvement at the left boundary yields a fourth-order convergence rate without further implementation of mesh refinement, Rannacher time-stepping, and/or smoothing of the initial condition. Furthermore, we extensively compare, the performance of our present method with several 5(4) Runge-Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine 5(4) pairs are faster and provide more accurate numerical solutions. Based on numerical results and comparison with other existing methods, we can validate that the present method is very fast and provides more accurate solutions with very coarse grids.
翻译:在用或不用红利解决美国选项时,数字方法往往获得较低的趋同率,如果即使采用高阶办法,即使没有实施进一步处理,也得不到进一步的处理。在本条中,我们提出了一个解决自由边界选项的快速和明确的第四级契约计划。特别是,资产选项和选择灵敏度的早期演练特征是根据非线性PDE与固定边界相结合的计算,而固定边界则获得高排序分析近似值。此外,我们在左边界实行新的处理方法,采用第三级Robin边界条件。我们不是从分析近似中计算最佳练习边界,而是从基于左边边界线性关系的资产选项中获取。因此,可以实现高顺序趋同率。我们通过实例证实,左边界的改进可产生第四级趋同率,而无需进一步实施网状改进、Rannacher时间跨步和/或平滑动初始条件。此外,我们广泛比较了我们目前方法的绩效与若干5(4)Runge-Kut对配方,而不是从分析近界关系中计算出最佳的界限,我们只是从资产选项选项中获取的。因此,可以实现高排序趋一致的趋一致的趋一致。我们用更精确的方法可以提供更精确的计算方法。