Extended Caputo space-fractional Black-Scholes equation with scale-dependent diffusion

Wannika Sawangtong; Doungporn Wiwatanapataphee; Panumart Sawangtong; Wannika Sawangtong; Doungporn Wiwatanapataphee; Panumart Sawangtong

doi:10.3934/math.2026306

AIMS Mathematics

2026, Volume 11, Issue 3: 7468-7496. doi: 10.3934/math.2026306

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Extended Caputo space-fractional Black-Scholes equation with scale-dependent diffusion

1.
Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
2.
Centre of Excellence in Mathematics, MHESI, Bangkok, 10400, Thailand
3.
Research group for fractional calculus theory and applications, Science and Technology Research Institute, King Mongkut's University of Technology North Bangkok, Bangkok, 10800, Thailand
4.
School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, WA 6845, Australia
5.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand

Received: 23 December 2025 Revised: 02 March 2026 Accepted: 10 March 2026 Published: 23 March 2026
MSC : 34K37, 35A22, 65R10

This paper developed an analytical framework for a space-fractional Black-Scholes model formulated with the extended Caputo fractional derivative. Fundamental operational properties of the extended Mellin integral transform, including shift rules, transform formulas for Caputo-type derivatives of orders $ 0 < \alpha\leq1 $ and $ 1 < \beta\leq2 $, and a convolution theorem, were established and used to treat scale-invariant fractional differential equations. By applying the extended Mellin transform to the governing Cauchy problem, we derived an explicit integral representation of the solution involving a gamma-function-based time-evolution multiplier. The validity of the representation was rigorously verified, and the classical Black-Scholes model with dividends was recovered as a special case. The model was applied to European put options, with numerical results validating the method and illustrating the impact of fractional dynamics. Calibration to SPY option market data demonstrates that the fractional parameters $ \alpha $ and $ \rho $ enhance flexibility in fitting observed option prices and capturing market-dependent scaling effects.
Citation: Wannika Sawangtong, Doungporn Wiwatanapataphee, Panumart Sawangtong. Extended Caputo space-fractional Black-Scholes equation with scale-dependent diffusion[J]. AIMS Mathematics, 2026, 11(3): 7468-7496. doi: 10.3934/math.2026306

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Abstract

This paper developed an analytical framework for a space-fractional Black-Scholes model formulated with the extended Caputo fractional derivative. Fundamental operational properties of the extended Mellin integral transform, including shift rules, transform formulas for Caputo-type derivatives of orders $ 0 < \alpha\leq1 $ and $ 1 < \beta\leq2 $, and a convolution theorem, were established and used to treat scale-invariant fractional differential equations. By applying the extended Mellin transform to the governing Cauchy problem, we derived an explicit integral representation of the solution involving a gamma-function-based time-evolution multiplier. The validity of the representation was rigorously verified, and the classical Black-Scholes model with dividends was recovered as a special case. The model was applied to European put options, with numerical results validating the method and illustrating the impact of fractional dynamics. Calibration to SPY option market data demonstrates that the fractional parameters $ \alpha $ and $ \rho $ enhance flexibility in fitting observed option prices and capturing market-dependent scaling effects.

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