This paper focuses on a scenario where a group of independently-operated investor-owned storage units seek to offer both energy and reserve in the day-ahead as well as the hour-ahead markets. We are particularly interested in the case when a significant portion of the power generated in the grid is from wind and other intermittent renewable energy sources. In this regard, we formulate a stochastic programming framework to choose optimal energy and reserve bids for the storage units. Our design takes into account the fluctuating nature of the market prices due to the randomness in the renewable power generation availability. We show that the formulated stochastic program can be converted into a convex optimization problem and therefore it can be solved efficiently. Our simulation results show that our design can assure profitability of the private investment on storage units. In particular, our design results in much higher profit compared to a similar but deterministic design that simply uses the expected values of the price parameters.