This paper investigates the pricing of American options under the finite moment log-stable
(FMLS) model. Under the FMLS model, the price of American-style options is governed
by a highly nonlinear fractional partial differential equation (FPDE) system, which is
much more complicated to solve than the corresponding Black–Scholes (B–S) system, with
difficulties arising from the semi-globalness of the fractional operator, in conjunction with
the nonlinearity associated with the early exercise nature of American-style options. Albeit
difficult, in this paper, we propose a new predictor–corrector scheme based on the spectral-
collocation method to solve for the prices of American options under the FMLS model. In
the current approach, the nonlinearity of the pricing system is successfully dealt with using
the predictor–corrector framework, whereas the non-localness of the fractional operator
is elegantly handled. We have also provided an elegant error analysis for the current
approach. Various numerical experiments suggest that the current method is fast and
efficient, and can be easily extended to price American-style options under other fractional
diffusion models. Based on the numerical results, we have also examined quantitatively the
influence of the tail index on American put options.