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

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 $0<\alpha \le 1$ and $1<\beta \le 2$, 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 $\alpha$ and $\rho$ enhance flexibility in fitting observed option prices and capturing market-dependent scaling effects.