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Energy storage systems (ESS) are considered as a promising solution to improve power system flexibility and facilitate the integration of renewables in electricity markets. This paper investigates the impact of strategic offering by an ESS operator in the day-ahead and balancing market. The offering strategy of a price-maker ESS operator is formulated as a bilevel model, where the upper-level problem represents the profit maximization of the ESS operator and the lower-level problem simulates the market-clearing outcome. This methodological framework can be used either to assess market efficiency distortion or as a trading strategy from the perspective of the ESS operator. Our analysis shows that adopting strategic behavior may improve ESS expected profit but reduces social welfare, especially for high ESS energy-to-power ratios.