On modified Mellin transforms, Gauss–Laguerre quadrature, and the valuation of American call options

We extend a framework based on Mellin transforms and show how to modify the approach to value American call options on dividend-paying stocks. We present a new integral equation to determine the price of an American call option and its free boundary using modified Mellin transforms. We also show how to derive the pricing formula for perpetual American call options using the new framework. A result due to Kim (1990) [24] regarding the optimal exercise price at expiry is also recovered. Finally, we apply Gauss–Laguerre quadrature for the purpose of an efficient and accurate numerical valuation.