In this study, we use a new approach, known as the Aboodh residual power series method (ARPSM), in order to obtain the analytical results of the Black–Scholes differential equations (BSDEs), which are prime for judgment of European call and put options on a non-dividend-paying stock, especially when they consist of time-fractional derivatives. The fractional derivative is considered in the Caputo sense. This approach is a combination of the Aboodh transform and the residual power series method (RPSM). The suggested approach is based on a new version of Taylor's series that generates a convergent series as a solution. The advantage of our strategy is that we can use the Aboodh transform operator to transform the fractional differential equation into an algebraic equation, which decreases the amount of computation required to obtain the solution in a subsequent algebraic step. The primary aspect of the proposed approach is how easily it computes the coefficients of terms in a series solution using the simple limit at infinity concept. In the RPSM, unknown coefficients in series solutions must be determined using the fractional derivative, and other well-known approximate analytical approaches like variational iteration, Adomian decomposition, and homotopy perturbation require the integration operators, which is challenging in the fractional case. Moreover, this approach solves problems without the need for He's polynomials and Adomian polynomials, so the small size of computation is the strength of this approach, which is an advantage over various series solution methods. The efficiency of the suggested approach is verified by results in graphs and numerical data. The recurrence errors at various levels of the fractional derivative are utilized to demonstrate the convergence evidence for the approximative solution to the exact solution. The comparison study is established in terms of the absolute errors of the approximate and exact solutions. We come to the conclusion that our approach is simple to apply and accurate based on the findings. [ABSTRACT FROM AUTHOR]