The usage of Lévy processes involving big moves or jumps over a short period of time has proven to be a successful strategy in financial analysis to capture such rare or extreme events of stock price dynamics. Models that follow the Lévy process are FMLS, Kobol, and CGMY models. Such simulations steadily raise the attention of researchers in science because of the certain best options they produce. Thus, the issue of resolving these three separate styles has gained more interest. In the new paper, we introduce the computational method of such models. At first, the left and right tempered fractional derivative with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the third kind (SCPTK). In the second point, by implementing finite difference approximation, we get the semi-discrete structure to solve the tempered fractional B–S model (TFBSM). We show that this system is stable and [Formula: see text] is the convergence order. In practice, the processing time and the calculation time per iteration will be reduced by a quickly stabilized system. Then we use SCPTK to approximate the spatial fractional derivative to get the full design. Finally, two numerical examples are provided to illustrate the established system’s reliability and effectiveness.

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The path of the Lévy process can be considered for prices of options such as a Rainbow or Basket option on two assets which leads to a 2D Black–Scholes model. The generalized model of this type of equation can be referred to as a 2D spatial-fractional Black–Scholes equation. The analytical solution of this kind is very complex and difficult and can even be said to be unattainable. For this reason, a numerical method has been proposed to solve it via the collocation method based on the Chebyshev orthogonal basis. Moreover, based on the derivatives in the called model, we approximated the derivative operator by using this type of base. Then we first obtained the temporal discrete form and finally the full-discrete form and turned it into a system of linear equations with the help of Chebyshev base roots.

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