Tension Leg Platforms (TLPs) are popular deep water platforms for extracting oil and gas. TLPs are connected to the sea bed by pretensioned tethers. If fluctuating tension in tethers is sizable compared to the pretension, many dynamic stability problems may arise which also affects the platform motion. In this paper, stochastic stability of TLP tethers in transverse oscillations is investigated using stochastic averaging technique and the probability flow of the reduced Fokker-Plank equation. Stochastic averaging is used to obtain drift and diffusion coefficients independent of time. Onset of such transverse oscillations may be caused due to regular or random vortex shedding. The tension fluctuation of the tether due to random vertical wave forces on the platform induces multiplicative random excitation in the tether because of the presence of nonlinear drag. As a consequence, dynamic instability may become a significant problem. The tether is modeled as a simply supported uniform beam under initial tension assuming transverse oscillation to take place only in its fundamental natural mode. Thus, the problem is reduced to a single degree of freedom system. Fluid resistance is calculated by Morison's equation consisting of drag and inertia forces. A semi-analytical procedure combined with stochastic averaging technique is employed to obtain the drift and diffusion coefficients of the response in closed form. By examining the probability flow in terms of drift and diffusion coefficients at the two boundaries, the stability of the system is investigated. Stability of tethers of a TLP in 300 m water depth, for different pretensions, is studied for sea states of 21 m, 16 m and 8 m significant wave heights represented by PM spectrum. It is shown that for low values of pretension, stochastic stability of tether in transverse direction may take place for large wave heights. © 2016 Elsevier Ltd. All rights reserved.