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It remains challenging to deploy existing risk-averse approaches to real-world applications. The reasons are multi-fold, including the lack of global optimality guarantee and the necessity of learning from long-term consecutive trajectories. Long-term consecutive trajectories are prone to involving visiting hazardous states, which is a major concern in the risk-averse setting. This paper proposes Short-Term VOlatility-controlled Policy Search (STOPS), a novel algorithm that solves risk-averse problems by learning from short-term trajectories instead of long-term trajectories. Short-term trajectories are more flexible to generate, and can avoid the danger of hazardous state visitations. By using an actor-critic scheme with an overparameterized two-layer neural network, our algorithm finds a globally optimal policy at a sublinear rate with proximal policy optimization and natural policy gradient, with effectiveness comparable to the state-of-the-art convergence rate of risk-neutral policy-search methods. The algorithm is evaluated on challenging Mujoco robot simulation tasks under the mean-variance evaluation metric. Both theoretical analysis and experimental results demonstrate a state-of-the-art level of STOPS' performance among existing risk-averse policy search methods.