时间分数阶Black-Scholes方程的L2紧致差分方法
L2 Compact Difference Method for Time Fractional Black-Scholes Equation
高毅 1吴立飞1
作者信息
- 1. 华北电力大学数理学院，北京 102206
- 折叠
摘要
针对经典Black-Scholes模型无法刻画金融市场长期记忆性的不足，研究时间分数阶Black-Scholes期权定价方程的高精度数值方法。基于Caputo分数阶导数的L2-1σ逼近和空间四阶紧致差分算子，构造了一种求解时间分数阶Black-Scholes方程的紧致差分格式。该格式在时间方向具有 阶精度，空间方向具有四阶精度。利用系数矩阵的严格对角占优性证明了解的存在唯一性；基于离散最大值原理和L2系数的有界性，证明了格式对任意步长均无条件稳定；通过截断误差估计和数学归纳法，证明了格式在离散最大模意义下以 阶收敛。数值实验验证了理论结果：空间收敛阶约为4.06，时间收敛阶约为2.00，纯时间ODE测试收敛阶约为2.44。期权定价实验表明，分数阶模型计算的期权价格高于经典模型， 越小记忆性越强、风险溢价越高。
Abstract
Aiming at the limitation that the classical Black-Scholes model cannot describe the long memory property of financial markets, a high-order numerical method for the time fractional Black-Scholes option pricing equation is studied. Based on the L2-1σ approximation of the Caputo fractional derivative and a fourth-order compact difference operator in space, a compact difference scheme for the time fractional Black-Scholes equation is constructed. The scheme achieves ( )-order accuracy in time and fourth-order accuracy in space. The existence and uniqueness of the solution are proved by the strict diagonal dominance of the coefficient matrix. Based on the discrete maximum principle and the boundedness of L2 coefficients, the scheme is proved to be unconditionally stable for arbitrary step sizes. By truncation error estimation and mathematical induction, the scheme is proved to converge at the order of in the discrete maximum norm. Numerical experiments verify the theoretical results: the spatial convergence order is about 4.06, the temporal convergence order is about 2.00, and the pure temporal ODE test yields a convergence order of about 2.44. Option pricing experiments show that the option prices computed by the fractional model are higher than those of the classical model, and a smaller leads to stronger memory and higher risk premium.关键词
计算数学/时间分数阶Black-Scholes方程/L2-1σ逼近/紧致差分格式/无条件稳定性Key words
computational mathematics/time fractional Black-Scholes equation/L2-1σ approximation/compact difference scheme/unconditional stability引用本文复制引用
高毅,吴立飞.时间分数阶Black-Scholes方程的L2紧致差分方法[EB/OL].(2026-07-20)[2026-07-23].http://www.paper.edu.cn/releasepaper/content/202607-21.学科分类
数学
