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研究求解固定利率抵押贷款模型的基于自适应网格的有限差分策略.采用中心差分格式来离散微分算子的空间变量导数项,构造分片一致的自适应网格,使得与离散算子相应的系数矩阵为M-阵,以保证所构造的差分策略是在无穷模意义下稳定的.通过区分不同网格点集,在相应的网格点集上应用极大模原理来直接导出误差估计.此有限差分策略对于任意波动率和任意利率都是稳定的,并且是关于标的资产价格二阶收敛的.数值实验证实了理论结果的准确性.
This paper studies the finite difference method based on adaptive grid which solves the fixed-rate mortgage loan model, and uses the central difference scheme to discretize the space variable derivative term of the differential operator to construct a piecewise adaptive grid so that the corresponding discrete operator The coefficient matrices are M-matrices to ensure that the constructed differential strategy is stable in the sense of infinite modulus. By distinguishing different sets of grid points, the maximum error principle is applied to the corresponding set of grid points to derive the error estimate directly. The finite difference strategy is stable for arbitrary volatility and arbitrary interest rate, and converges secondarily for underlying asset prices. Numerical experiments confirm the accuracy of theoretical results.