This paper proposes a framework based on the celebrated transform of Mellin type (MT) for the direct solution of the Black-Scholes-Merton European Put Option Model (BSMEPOM) on Dividend Yield (DY) with Modified-Log Payoff Function (MLPF) under the geometric Brownian motion. The focal goal of this paper is to use MT to obtain a valuation formula for the European Put Option (EPO) which pays a DY with MLPF. By means of the MT and its inversion formula, the price of EPO on DY was expressed in terms of integral equation. The valuation formula of EPO was obtained with the help of the convolution property of MT and final time condition. MT was tested on an illustrative example in order to measure its performance, effectiveness and suitability. The MLPF was compared with other existing payoff functions. Hence, the effect of DY on the pricing of EPO with MLPF was also investigated.

This paper proposes a framework based on the celebrated transform of Mellin type (MT) for the analytic solution of the Black-Scholes-Merton European Power Put Option Model (BSMEPPOM) on Dividend Yield (DY) with Modified-Log-Power Payoff Function (MLPPF) under the geometric Brownian motion. The MT has the capability of tackling complex functions by means of its fundamental properties and it is closely related to other well-known transforms such as Laplace and Fourier types. The main goal of this paper is to use MT to obtain a valuation formula for the European Power Put Option (EPPO) which pays a DY with MLPPF. By means of MT and its inversion formula, the price of EPPO on DY was expressed in terms of integral equation. Moreover, the valuation formula of EPPO was obtained with the help of the convolution property of MT and final time condition. The MT was tested on an illustrative example in order to measure its performance, effectiveness and suitability. The MLPPF was compared with other existing payoff functions. Hence, the effect of DY on the pricing of EPPO with MLPPF was also investigated.