分数阶Black-Scholes模型下的波动率校准问题  

Volatility Calibration Problems in A Fractional Black-Scholes Models

作　　者：杨培 许作良 YANG Pei;XU Zuoliang(College of Statistics and Mathematics,Hebei University of Economics and Business,Shijiazhuang 050061,China;School of Mathematics,Renmin University of China,Beijing 100872,China)

机构地区：[1]河北经贸大学统计与数学学院,河北石家庄050061 [2]中国人民大学数学学院,北京100872

出　　处：《宁夏大学学报(自然科学版中英文)》2025年第1期24-33,共10页Journal of Ningxia University(Natural Science Edition)

基　　金：国家自然科学基金资助项目(12071479)。

摘　　要：对分数阶Black-Scholes模型中的波动率函数进行反演,提出了一种准确、稳健的数值算法.首先,对于正问题,考虑到收益函数的奇异性会影响L1格式的收敛速度,提出了一种基于改进L1格式的有限差分方法.此数值方法能有效恢复L1格式的收敛性,且在计算中只需求解稀疏的三对角线性系统.其次,对于反问题,考虑了随时间变化的波动率函数,波动率反演问题可以表述为求解损失函数的最小值.构造了一个连续、分段线性的波动率函数,并采用预测-校正方法来抑制可能的振荡.数值计算和实证分析的结果表明了所提方法的准确性和可靠性.In this paper,an accurate and robust numerical algorithm is proposed to invert the volatility function in the fractional Black-Scholes model.First,for the direct problem,considering that the singularity of the payoff function affects the convergence speed of the L1 method,a finite difference method based on the improved L1 method is proposed.This numerical method can effectively recover the convergence of the L1 method,and only sparse tridiagonal linear systems need to be solved during the computation.Moreover,for the inverse problem,considering the time-dependent volatility function,the volatility inversion problem can be formulated as minimizing the loss function.A continuous and piecewise linear volatility function is constructed and a predictor-corrector approach to mitigate potential oscillations is employed.The results of numerical simulations and empirical analyses demonstrate the accuracy and reliability of the proposed method.

关 键 词：分数阶BS模型 反问题 改进L1格式 欧式期权 波动率校准 

分 类 号：O242[理学—计算数学] F830.9[理学—数学]