双币种期权定价模型的一个新ADI并行差分方法

A New ADI Parallel Difference Method for Quanto Options Pricing Model

双币种期权是一种重要的金融衍生产品,其定价模型是一个含有混合导数项的二维Black-Scholes方程,研究它的数值解法有着非常重要的理论意义和实际价值.本文给出求解双币种期权定价模型的基于Craig-Sneyd分裂法的一个新ADI差分方法(C-S ADI),该方法首先将二维B1ack-Scholes方程分裂为两个一维方程和一个含有混合导数的二维方程,然后分别对一维方程构造半隐式格式,对含混合导数的二维方程构造显式格式进行计算.C-S ADI差分方法具有以下优点:并行性,无条件稳定性,收敛性及空间二阶、时间一阶的计算精度.理论分析与数值试验表明,相比于经典的Crank-Nicolson差分格式和已有的基于Douglas Rachford分裂法的ADI差分格式(D-R ADI),本文格式计算精度更高,并且由于其具有天然的并行特性,本文格式比串行的Crank-Nicolson格式节省了近1/5的计算时间,证实了该方法对求解双币种期权定价模型是有效的.

Quanto options is a very important financial derivative, and its pricing model is a two-dimensional Black-Scholes equation with a mixed derivative term. The research of the equation＇s numerical solution is value both in theory and practice. This paper gives a new ADI difference method based on the Craig-Sneyd splitting technique （C-S ADI） for solving the quanto options pricing model. This C-S AI）I method first splits the two-dimensional B-S equation into two separate one-dimensional problems and one two-dimensional problem with a mixed term. And then solve them by semi-implicit difference scheme and explicit difference scheme per time-step respectively to get the solution. The C-S ADI method has the following advantages： parallelism, unconditional stability, convergency, and the calculation of second-order in space, first-order in time. The numerical experiments show this method is very efficient and gives better accu- racy than the existent Crank-Nicolson difference scheme and the ADI method based on the Douglas-Rachford splitting technique （D-R ADI）. What＇s more, as the natural parallel property of the C-S ADI method, it is easy to realize parallel computing, and the saving calculation time of it is nearly 1/5 of the serial Crank-Nicolson scheme＇s. Thus the method given by this paper can be used to solve the quanto options pricing problems effectively.