A simple and fast method for valuing American knock-out options with rebates

In this paper, we derive an analytic formula for the American knock-out options with rebate. Rather than using a probabilistic method, we use the Laplace-Carson Transform(LCT) method to induce a simple functional equation associated with the complex problem of option pricing Partial Differential equation with free boundary. The transformed value of free boundary could be solved by applying Newton's method. Lastly, numerical Laplace inversion techniques are used to solve for the wanted free boundary value and the options value.