Approximate price of the option under discretization by applying fractional quadratic interpolation

The time-fractional Black-Scholes model (TFBSM) governing European options in which the temporal derivative is focused on the Caputo fractional derivative with ۰ < β ≤ ۱ is considered in this article. Approximating financial options with respect to their hereditary characteristics can be well understood and explained due to its outstanding memory effect current in fractional derivatives. Compelled by the stated cause, It is important to find reasonably accurate and successful numerical methods when approaching fractional differential equations. The simulation model given here is developed in two ways: one, the semi-discrete is produced in the time using a quadratic interpolation with the order of precision τ۳−α in the case of a smooth solution, and subsequently, the unconditional stability and convergence order are investigated. The spatial derivative variables are simulated using the collocation approach based on a Legendre basis for the designed full-discrete scheme. Last, we employ various test problems to demonstrate the suggested design’s high precision. Moreover, the obtained results are compared to those obtained using other methodologies, demonstrating that the proposed technique is highly accurate and practicable.

Keywords:

Time fractional Black-Scholes model , Square interpolation , Legendre polynomials , Collocation method