Closed-form solution for the critical stock price and the price of perpetual American call options via the improved Mellin transforms

Abstract

This paper presents the closed-form solution for the critical stock price 'also known as the free boundary' and the price of the perpetual American call options by means of the improved Mellin transforms. The expression for the price of the perpetual American call options with non-dividend yield is obtained as a steady-state solution to the non-homogeneous Black-Scholes partial differential equation for the American call options, rather than as a solution to the 'static' problem. The result shows that the critical stock price for the American call option is infinite and the price of the option coincides with the underlying asset price. For better accuracy, we have shown that our integral representation for the price of the American call option can be evaluated by means of an N-point Gauss-Laguerre quadrature method. To determine the performance of the improved Mellin transforms, we have compared the results generated by the improved Mellin transforms with the binomial model and the Black-Scholes model for the valuation of the American call option with non-dividend yield. The numerical results show that the improved Mellin transforms agrees with the Black-Scholes model and performs better than the binomial model as shown in Figure 3.