In this paper

we investigate the European up and out call option under the framework of finite moment log-stable process (FMLS). In this case

the pricing model is boundary problem with fractional partial differential equation and it is impossible obtain the analytical solution of this mathematical model

so that the numerical method is considered. A discrete full implicit scheme with second order accuracy should be constructed. Moreover

in order to improve the computational efficiency

the preconditioned conjugate gradient normal residual (PCGNR) method with a circulant precondition is employed to solve the linear systems in this paper

and the fast Fourier transform (FFT) can be utilized to reduce the computational cost of the matrix-vector multiplication. In the end

we analyse the impacts of key parameters in our model to numerical results.