Extended Caputo space-fractional Black-Scholes equation with scale-dependent diffusion
Issued Date
2026-01-01
Resource Type
eISSN
24736988
Scopus ID
2-s2.0-105033762044
Journal Title
Aims Mathematics
Volume
11
Issue
3
Start Page
7468
End Page
7496
Rights Holder(s)
SCOPUS
Bibliographic Citation
Aims Mathematics Vol.11 No.3 (2026) , 7468-7496
Suggested Citation
Sawangtong W., Wiwatanapataphee D., Sawangtong P. Extended Caputo space-fractional Black-Scholes equation with scale-dependent diffusion. Aims Mathematics Vol.11 No.3 (2026) , 7468-7496. 7496. doi:10.3934/math.2026306 Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/115959
Title
Extended Caputo space-fractional Black-Scholes equation with scale-dependent diffusion
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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 < α ≤ 1 and 1 < β ≤ 2, 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 α and ρ enhance flexibility in fitting observed option prices and capturing market-dependent scaling effects.