The Convergence Investigation of a Numerical Scheme for the Tempered Fractional Black-Scholes Model Arising European Double Barrier Option

The application of Lévy processes including major movements or jumps over a small period of time has proved to be an effective technique in financial research to catch certain unusual or extreme cases in stock price dynamics. Models that follow the Lévy process are The FMLS, Kobol, and CGMY models. These models gradually grow the interest for study among researchers because of some of the best choices them. Therefore the topic of approaching these three different models has drawn yet more attention. In the current paper, we present these models’ numerical method. At first, The Riemann-Liouville tempered fractional derivative (RLTFD) with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the fourth kind (SCPFK). In the second step, we gain the semi-discrete design to solve the tempered fractional B-S model (TFBSM) by applying finite difference approximation. We’re going to show that this system is stable and $$\mathcal {O}(\delta \tau )$$ O ( δ τ ) is the convergence order. In fact, a fast stabilized method will obtain to reduce the time from processing and the computation time per repetition. Then to get the full scheme, we use SCPFK to approximate the spatial fractional derivative. Finally, two numerical examples are presented to demonstrate the accuracy and usefulness of the developed system.