Cognitive radio with spectrum sharing feature is a promising technique to address the spectrum under-utilization problem in dynamically changing environments. In this paper, achievable capacity gain of spectrum sharing systems over dynamic fading environments is studied. For the analysis, a theoretical fading model called hyper fading model that is suitable to the dynamic nature of cognitive radio channel is proposed. Closed-form expression of probability density function (PDF) and cumulative density function (CDF) of the signal-to-noise ratio (SNR) for secondary users in spectrum sharing systems are derived. In addition, the capacity gains achievable with spectrum sharing systems in high and low power regions are obtained. Numerical simulations are performed to study the effects of different fading figures, average powers, interference temperature, and number of secondary users on the achievable capacity.