美式分期付款期权定价模型的有限差分法

A Finite Di?erence Scheme for Pricing American Continuous-installment Options

由于Black-Scholes微分算子是对流占主的微分算子,对其在等距网格上应用中心差分格式离散会导致数值解产生非物理震荡.本文对连续支付美式分期付款期权定价模型构造了基于自适应网格的有限差分策略,它采用中心差分格式离散空间变量导数项,构造分片一致网格使得与离散算子相应的系数矩阵为M-阵,以保证所构造差分策略对于任意波动率和任意利率都是无穷模意义下稳定的.应用光滑化技巧来有效处理终值条件的不光滑性,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,证得此有限差分策略是关于标的资产价格二阶收敛的,并且利用数值解求得美式分期付款期权的最优执行边界和最优终止边界,数值实验证实了理论结果的准确性.

Since the Black-Scholes partial differential operator is a convection-dominated op- erator, the central difference scheme on a uniform mesh may produce nonphysical oscillations in the numerical solution. In this paper, a stable numerical method for the linear complementary problem arising from American continuous-installment option pricing is presented. The numer- ical method is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique. The matrix associated with the discrete operator is an M-matrix, which ensures that the scheme is maximum-norm stable for arbitrary volatility and arbitrary interest rate. A smoothing technique is used to treat the non-smoothness of the payoff function. The maximum principle is applied to the discrete linear complementary problem in two mesh sets and the error estimates are derived. It is proved that the scheme is second-order convergent with respect to the spatial variable. From the numerical solution, the optimal stopping and exercise boundaries axe obtained. Numerical experiments, to demonstrate the effectiveness of the proposed method, show that the approach is stable and accurate.