基于分数阶Black-Scholes方程中的隐含波动率反问题  

The inverse problem of implied volatility with source term in the fractional Black-Scholes equation

作　　者：宋苗苗 SONG Miaomiao(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学数理学院,甘肃兰州730070

出　　处：《商丘师范学院学报》2025年第3期20-25,共6页Journal of Shangqiu Normal University

基　　金：国家自然科学基金资助项目(61663018,11961042);甘肃省自然科学基金资助项目(22JR5RA341)。

摘　　要：研究了分数阶Black-Scholes方程(TFBSM)中的双障碍期权及时间相关隐含波动率系数的恢复.与传统的双障碍期权模型不同,模型TFBSM引入了源项,以更好的模拟市场波动性的变化,更准确地定价双障碍期权.研究包括两个方面:一方面是针对正向问题,即采用了L1-中心差分隐式逼近方案,有效地解决了初始边界值问题,同时提供了正向问题求解的收敛性分析.另一方面是逆问题,采用线性化技术,将直接问题转化为一个逆源问题.通过额外的数据来恢复隐含波动率,并证明了唯一性.This paper investigates the recovery of the time-dependent implied volatility coefficient for double-barrier options in the context of the fractional Black-Scholes equation(TFBSM).In contrast to traditional double-barrier option models,our TFBSM model incorporates a source term to better capture changes in market volatility,resulting in a more accurate pricing of double-barrier options.The study in this paper comprises two main aspects.Firstly,for the forward problem,we employ an LI-Central Difference Implicit Approximation(LI-CDIA)scheme to effectively solve the initial boundary value problems,and provide a convergence analysis for the forward problem solver.Secondly,for the inverse problem,we utilize linearization techniques to transform the direct problem into an inverse source problem.This allows for the recovery of implied volatility using additional data,and uniqueness is demonstrated in this context.

关 键 词：反问题 时间分数阶偏微分方程 隐含波动率 唯一性 

分 类 号：O175.26[理学—数学]