This paper proposes a model of the behavior of an expected profit-maximizing merchant storage owner with the ability to exercise unilateral market power. The resulting non-linear bilevel optimization problem is transformed into a single-level stochastic bilinear program using the Karush-Kuhn-Tucker conditions of the lower-level Independent System Operator dispatch problem. By discretizing the offers and bids of the merchant storage owner, the problem is formulated as a stochastic disjunctive program. Using the disjunctive nature of the derived program, a specialized branch-and-bound algorithm that applies a linear quasi-relaxation of the merchant storage problem is proposed. Our solution algorithm is able to solve the problem in an efficient manner; returning the charge and discharge strategies for the merchant storage owner that yield the highest expected profits. Simulations of test systems reveal the various abilities of the merchant storage owner to exercise unilateral market power. Those include demand withholding, generation withholding and under-usewhich result in an increased congestion in both space and time when compared to the welfare-maximizing use of storage. Factors such as uncertain bids by other players, final state-of-charge requirements and arbitrage by other storage players are investigated. Moreover, numerical results demonstrate the superior computational performance of the proposed solution algorithm when benchmarked against current practices in the literature.