Energy storage systems (ESS) may provide the required flexibility to cost-effectively integrate stochastic electricity generation from renewable energy sources. Energy storage owners should, however, be properly renumerated for the services they provide. In this paper, we propose a bilevel optimization problem, mimicking the strategic behavior of a price-making ESS owner in a joint day-ahead (DA) energy-reserve market. We explicitly consider possible real-time balancing market outcomes when clearing the DA markets via probabilistic reserve constraints and ensure the feasibility of dispatching ESS-based reserves through novel worst-case reserve activation constraints. The proposed framework may be used by regulators and market operators to assess market design options or by ESS owners to optimize their bids.