LAUSR

In this paper, a free boundary fractional Black‐Scholes (FBS) model of American put option pricing is investigated. To convert the free boundary FBS model to a model with known boundary condition, the quasi‐stationary method is applied, which leads to solvability of the American put option problem. Then, a nonstandard finite difference method and Grünwald‐Letnikov approximation are respectively used to approximate the derivatives with respect to stock price and time fractional derivative to get a fractional nonstandard finite difference problem. It is then shown that the proposed method is stable and convergent. The uniqueness of the approximate solution is also proved for the proposed method. Our numerical results indicate that the general physical conditions of American put option valuation formulas under the FBS model are satisfied.