固定利率抵押贷款模型的基于自适应网格的有限差分法

A FINITE DIFFERENCE SCHEME BASED ON A LAYER-ADAPTIVE MESH FOR THE FIXED RATE MORTGAGES MODEL

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.

In this paper we adopt the adaptive mesh approach to solve the model for pricing the fixed-rate mortgage, which is a convection-dominated problem when the volatility or the asset price is small. It is well known that when using the standard finite difference method to solve those convection-dominated problems, numerical dif- ficulty can be caused. Our numerical method combines a central difference spatial discretization on a piecewise uniform mesh with an implicit time stepping technique.The matrix associated with discrete operator is an M-matrix, which ensures that the scheme is stable for arbitrary volatility and arbitrary interest rate without any extra conditions. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. It is proved that the scheme is second order convergent with respect to the spatial variable. Numerical results in- dicate that the numerical solutions by our method are non-oscillatory and the scheme is second order convergent with respect to the spatial variable.