This paper examines the analytical option pricing of trades involving L\'evy exponential assets characterized by infinite activity, specifically focusing on the one-sided Tempered Stable model. This class of L\'evy process, which allows for an infinite number of jumps within any finite time interval, presents a more complex framework than other L\'evy models. The objective is to study a comprehensive and unified pricing framework for the one-sided Tempered Stable process utilizing the Mellin transform and residual calculus, and to calculate the price. The results show this method is reliable and effective.
