Uniqueness of Approximate Solution for American Put Option Pricing

We introduce the mathematical modeling of American put option under the Fractional Black- Scholes (FBS) model, which leads to a free boundary problem. Then the free boundary (optimal exercise boundary) that is unknown, is found by the quasi-stationary method that cause American put option problem to be solvable. In continuation we use a finite difference method for derivatives with respect to stock price, backward finite difference formula for derivatives with respect to time and reach a fractional finite difference problem. We show that the set up fractional finite difference problem has a unique solution.