On Modified Mellin Transforms, Gauss-Laguerre Quadrature and the Valuation of American Call Options

We extend a framework based on Mellin transform techniques and show how the approach can be modified to value American call options on dividend paying stocks. We derive a new integral equation to determine the price of an American call option and its free boundary using modified Mellin transforms. We show how the new framework can be used to derive the valuation formula for perpetual American call options and use the new integral characterization to recover a result due to Kim (1990) regarding the optimal exercise price of American call options at expiry. Finally, we apply Gauss-Laguerre quadrature for the purpose of an efficient and accurate American call option valuation