In this paper, a new algorithm is designed based on a state feedback global error control system, Laplace trans form, order reduction method, and k-step numerical integration methods to numerically solve the Black-Scholes equation. For this purpose, the Black-Scholes equation is converted into a first-order system of ordinary differ ential equations by using the Laplace transform and order reduction method. Also, a new robust linear optimal adaptive global error control dynamic for designing an adaptive time variant step size sequence is modeled and a corresponding optimal control law based on robust and optimal eigenvalue assignment is designed. The proposed optimal control law guarantees the absolute stability of the implemented k-step numerical integrator methods. Finally, the transformed approximate solution of the Black-Scholes equation has been obtained using the Stefhest inverse Laplace transformation algorithm. The simulation examples show that the optimal control of global error under a given tolerance level, the guarantee of absolute stability, and the best approximation of sensitivity analysis indexes for the proposed approximate solution of the Black-Scholes equation is among the important advantages of the proposed method.
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Alishahi, M. and Yarahmadi, M. (2026). New optimal adaptive stepsize algorithm for solving black-scholes equation. Computational Methods for Differential Equations, 14(2), 633-651. doi: 10.22034/cmde.2025.65090.2974
MLA
Alishahi, M. , and Yarahmadi, M. . "New optimal adaptive stepsize algorithm for solving black-scholes equation", Computational Methods for Differential Equations, 14, 2, 2026, 633-651. doi: 10.22034/cmde.2025.65090.2974
HARVARD
Alishahi, M., Yarahmadi, M. (2026). 'New optimal adaptive stepsize algorithm for solving black-scholes equation', Computational Methods for Differential Equations, 14(2), pp. 633-651. doi: 10.22034/cmde.2025.65090.2974
CHICAGO
M. Alishahi and M. Yarahmadi, "New optimal adaptive stepsize algorithm for solving black-scholes equation," Computational Methods for Differential Equations, 14 2 (2026): 633-651, doi: 10.22034/cmde.2025.65090.2974
VANCOUVER
Alishahi, M., Yarahmadi, M. New optimal adaptive stepsize algorithm for solving black-scholes equation. Computational Methods for Differential Equations, 2026; 14(2): 633-651. doi: 10.22034/cmde.2025.65090.2974
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