Numerical investigation of the time-fractional Black-Scholes equation with barrier choice of regulating European option

The price variance of the associated fractal transmission mechanism was used to estimate the Black-Scholes fractional model of which a time-fractional derivative is $alpha$. In the current paper, the time-fractional Black-Scholes equation (TFBSE) that the temporal derivative is the Caputo fractional derivative is known by regulating the European option. At first, linear interpolation with a temporally tau²⁻ᵃˡᵖʰᵃ order accuracy is used for constructing the semi-discrete. Then, the spatial derivative terms are approximated with the help of the collocation approach centered on the  Chebyshev polynomials of the third form (CPTF). Finally, The unconditional stability and convergence order are analyzed by applying the energy method. To show the precision of the numerical  system, we solved two instances of the TFBSE. Numerical results and comparisons indicate the proposed approach is very reliable and efficient.