This report presents two compact finite difference schemes and a collocation method on uniform mesh for solving the fractional Black-Scholes partial differential equation for the European type option. The time-fractional derivative is approximated by the $L1-2$ formula and the $L2-1_{\sigma }$ formula respectively, and two compact difference schemes with orders $O((\Delta t)^{3-\alpha} +(\Delta x)^4)$ and $O((\Delta t)^2 + (\Delta x)^4)$ are constructed. Also, we develop a new scheme by using the $L1-2$ formula to approximate the time-fractional derivative and the Laguerre polynomial to approximate the spatial derivatives. The stability and convergence analysis of all the schemes is also analyzed. Finally, numerical examples are carried out to verify the accuracy and effectiveness of all the schemes, and these schemes are compared.