  In this paper, a high-order and fast numerical method is investigated for the
time-fractional Black-Scholes equation. In order to deal with the typical weak
initial singularities of the solution, we construct a finite difference scheme
with variable time steps, where the fractional derivative is approximated by
the nonuniform Alikhanov formula and the sum-of-exponentials (SOE) technique.
In the spatial direction, an average approximation with fourth-order accuracy
is employed. The stability and the convergence with second-order in time and
fourth-order in space of the proposed scheme are religiously derived by the
energy method. Numerical examples are given to demonstrate the theoretical
statement.
