CAJ | 학술논문

时间分数阶期权定价模型(时间分数阶Black-Scholes方程)数值解法的研究具有重要的理论意义和实际应用价值。对时间分数阶Black-Scholes方程构造了显-隐格式和隐-显差分格式，讨论了两类格式解的存在唯一性，稳定性和收敛性。理论分析证实，显-隐格式和隐-显格式均为无条件稳定和收敛的，两种格式具有相同的计算量。数值试验表明：显-隐和隐-显格式的计算精度与经典Crank-Nicolson(C-N)格式的计算精度相当，其计算效率(计算时间)比C-N格式提高30%。数值试验验证了理论分析，表明本文的显-隐和隐-显差分方法对求解时间分数阶期权定价模型是高效的，证实了时间分数阶Black-Scholes方程更符合实际金融市场。
시간분수계기권정개모형(시간분수계Black-Scholes방정)수치해법적연구구유중요적이론의의화실제응용개치。대시간분수계Black-Scholes방정구조료현-은격식화은-현차분격식，토론료량류격식해적존재유일성，은정성화수렴성。이론분석증실，현-은격식화은-현격식균위무조건은정화수렴적，량충격식구유상동적계산량。수치시험표명：현-은화은-현격식적계산정도여경전Crank-Nicolson(C-N)격식적계산정도상당，기계산효솔(계산시간)비C-N격식제고30%。수치시험험증료이론분석，표명본문적현-은화은-현차분방법대구해시간분수계기권정개모형시고효적，증실료시간분수계Black-Scholes방정경부합실제금융시장。
It is important in the application to study the numerical computation for time-fractional option pricing model (time-fractional Black-Scholes equation). Explicit-Implicit (E-I) scheme and Implicit-Explicit (I-E) scheme are constructed for solving time-fractional Black-Scholes equation. The stable, convergent, existence and uniqueness of solutions given by these schemes are discussed. Theoretical analysis demonstrates that E-I and I-E schemes are unconditional stability and convergent. They have the same calculation. Numerical experiments show that computational accuracy of E-I and I-E schemes is similar to the classic Crank-Nicolson (C-N) scheme, and their computational effciency (computing time) is 30% higher than C-N scheme. Theoretical analysis and numerical experiments demonstrate the superiority of E-I and I-E schemes for solving time-fractional option pricing model, and affrm that time-fractional Black-Scholes equation is more in line with the actual financial market.