非线性Black-Scholes方程有限差分并行计算的新方法研究

In order to improve the computational efficiency in solving B-S (Black-Schloes) equation of option pricing problem, this project researches on new parallel difference method based on preconditioning JFNK (Jacobian-Free Newton-Krylov) method. It is a kind of fast algorithm which combined nonlinear outer circulation Newton iteration with linear inner circulation Krylov iteration. Its advantage is that the formation and storage of Jacobian matrix to Newton iteration is avoided. Its effectiveness depends on the preconditioning method to Krylov inner loop. The goals are to improve the computation speed of JFNK method and make sure stability of parallel schemes. Then the key to achieve the two goals is to establish Alternating Segment (Block) Explicit-Implicit or Pure Implicit scheme and find effective preconditioning JFNK method. Combining various Alternating Parallel Explicit-Implicit or Pure Implicit method with preconditioning JFNK method is the feature of this project. The innovation of this project is to construct effective preconditioning JFNK algorithm based on parallel difference scheme, which can take stability, accuracy and parallelism into account at the same time. Finally, this method is tested in multidimensional B-S equations, and it will make new contribution for the development and improvement of computational finance for our country.

为适应期权定价问题中B-S（Black-Scholes）方程快速求解的需要，本项目研究预处理JFNK（Jacobian-Free Newton-Krylov）方法结合并行差分格式求解B-S方程的新方法。这是一种非线性外循环Newton迭代与线性内循环Krylov迭代相结合的快速算法，其优点是进行Newton迭代时不要求Jacobian矩阵的形成和存储；JFNK方法的有效性取决于内循环中线性系统的预处理。提高JFNK方法的求解速度与保证差分格式的计算稳定性是项目的目标；建立交替分段（块）显-隐式或纯隐式格式与构造有效预处理JFNK方法是实现这两个目标的关键；将多种显-隐式或纯隐式交替并行方法纳入JFNK方法的构造之中是项目的特色；在并行差分格式的基础上构造JFNK方法使算法同时兼顾计算的稳定性、准确性和并行性是项目的创新点。在多维B-S方程中作检验，为我国计算金融学的发展与提高做出新贡献。

B-S（Black-Scholes）模型的数值解法对许多金融衍生品定价研究发挥着显著的促进作用，B-S方程的并行算法研究具有重要的科学意义和金融应用价值。对一维非线性B-S方程，基于显-隐交替方法的思想，提出交替分组并行差分方法，构造AGE、ASE-I、ASI-E和IASC-N并行差分格式，用JFNK方法求解ASE-I和ASC-N并行差分格式。基于分裂法求解二维B-S方程，提出交替分块（带）并行差分方法，构造ABdC-N 和D-R ADI、C-S ADI格式。对三维B-S方程，构造AOS-显隐格式和AOS-隐显格式；构造分数阶B-S方程的并行差分格式（ASE-I、ASI-E，PASE-I、PASI-E，MASC-N）。理论分析和数值试验均表明，这些差分格式是无条件稳定和收敛的，多数为二阶精度格式，计算方法具有并行性、省时性，其综合性能优于经典的串行差分格式，说明项目构造的并行差分格式求解B-S方程是有效的。