Option pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black–Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second‐order spatial derivative of the BS model is replaced by a fractional operator of order α which generates an α‐stable Lévy process. In this paper, we consider the finite difference method to approximate the FMLS model. We present two numerical schemes for this approximation: the implicit numerical scheme and the Crank–Nicolson scheme. We carry out convergence and stability analyses for the proposed schemes. Since the fractional operator routinely generates dense matrices which often require high computational cost and storage memory, we explore three methods for solving the approximation schemes: the Gaussian elimination method, the bi‐conjugate gradient stabilized method (Bi‐CGSTAB) and the fast Bi‐CGSTAB (FBi‐CGSTAB) in order to compare the cost of calculations. Finally, two numerical examples with exact solutions are presented where we also use extrapolation techniques to achieve higher‐order convergence. The results suggest that the proposed schemes are unconditionally stable and convergent, and the FMLS model is useful for pricing options.

中文翻译：

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欧式看涨期权有限矩对数稳定计算模型的新数值技术

期权定价模型通常用于描述金融市场价格的动态特征。与经典的Black-Scholes（BS）模型不同，有限力矩对数稳定（FMLS）模型可以解释小时间步长内的价格大变动。在FMLS中，BS模型的二阶空间导数由α阶的分数运算符代替，该运算符生成α稳定的Lévy流程。在本文中，我们考虑用有限差分法近似FMLS模型。我们提出了两种近似的数值方案：隐式数值方案和Crank-Nicolson方案。我们对所提出的方案进行收敛性和稳定性分析。由于分数运算符通常会生成密集矩阵，这些矩阵通常需要较高的计算成本和存储内存，因此我们探索了三种解决近似方案的方法：高斯消元法，双共轭梯度稳定方法（Bi-CGSTAB）和快速Bi- CGSTAB（FBi-CGSTAB），以便比较计算成本。最后，给出了两个具有精确解的数值示例，其中我们还使用外推技术来实现高阶收敛。