A Fully Implicit Numerical Model for Time-Fractional Multiasset Black–Scholes Equations

Yan Zhou and Yunxing Zhang

Journal of Applied Mathematics, 2026, vol. 2026, 1-11

Abstract: This paper proposes and validates a finite-difference numerical model for time-fractional multiasset Black–Scholes option-pricing problems. The model adopts a Caputo time derivative of order α∈0,1 to capture the memory effects of asset prices. A logarithmic coordinate transformation is first applied to convert the pricing equation into a constant-coefficient convection–diffusion–reaction equation. The temporal discretization is discretized using a weighted fully implicit L1-scheme, and the spatial difference operator is assembled as the tensor product of one-dimensional second-order central-difference operators. This formulation yields discrete systems with regular sparse structures in both two- and three-asset cases, so that a single sparse LU-factorization can be reused across all time levels. Under the assumption of smooth solutions, the global error satisfies OΔτ2−α+Oh2. Numerical experiments are conducted to validate the present model at two stages. Manufactured-solution tests confirm the expected temporal convergence of order 2−α and second-order spatial accuracy, with convergence rates independent of dimension. Point-wise comparisons are further performed for a one-asset European call option against the Laplace theoretical solution and for a two-asset exchange option against the Margrabe analytical solution, which verify the pricing accuracy for low-asset problems. As a practical application, the proposed model is employed to price a three-asset equally weighted basket call option. Sensitivity analysis demonstrates that the impact of the fractional order on option prices depends systematically on the moneyness of the payoff. The developed model therefore provides a deterministic, order-preserving, and cost-effective tool for pricing European, exchange, and multiasset basket options under market memory effects. It can be directly applied to price computation and parameter-sensitivity analysis.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:1906726

DOI: 10.1155/jama/1906726

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