Numerical pricing of European options under time-fractional Black-Scholes equation in financial markets.

The time-fractional Black-Scholes model (TFBSM) incorporating a Caputo derivative operator is developed to analyze price dynamics in fractal market systems with correlated fluctuations. This paper focuses on developing a numerical approach to derive an approximate solution for the TFBSM of order 0<α<1 in problems related to European options using the Legendre collocation technique. The primary advantage of this method is that the function is approximated with shifted Legendre polynomials in both space and time directions, using fewer nodal points for improved results. Moreover, a detailed discussion on the error analysis of the proposed method is provided. Finally, three numerical examples are presented to validate the precision of the proposed strategy.