  The Black-Scholes (B-S) equation has been recently extended as a kind of
tempered time-fractional B-S equations, which becomes an interesting
mathematical model in option pricing. In this study, we provide a fast
numerical method to approximate the solution of the tempered time-fractional
B-S model. To achieve high-order accuracy in space and overcome the weak
initial singularity of exact solution, we combine the compact difference
operator with L1-type approximation under nonuniform time steps to yield the
numerical scheme. The convergence of the proposed difference scheme is proved
to be unconditionally stable. Moreover, the kernel function in the tempered
Caputo fractional derivative is approximated by sum-of-exponentials, which
leads to a fast unconditionally stable compact difference method that reduces
the computational cost. Finally, numerical results demonstrate the
effectiveness of the proposed methods.
