This paper proposes a novel method to generate bid ceilings for energy storage in electricity markets to facilitate social welfare convergence and regulate potential market power exercises. We derive the bid bounds based on a tractable multi-period economic dispatch chance-constrained formulation that systematically incorporates the uncertainty and risk preference of the system operator. The key analytical results verify that the bounds effectively cap the truthful storage bid across all uncertainty scenarios with a guaranteed confidence level. And the cleared storage bids should be bounded by the risk-aware locational marginal price. We show that bid bonds decrease as the state of charge increases but rise with greater net load uncertainty and risk preference. We test the effectiveness of the proposed pricing mechanism based on the 8-bus ISO-NE test system, including agent-based storage bidding models. Simulation results show that the bid bounds effectively adjust storage bids to align with the social welfare objective. Under 30% renewable capacity and 20% storage capacity, the bid bounds contribute to an average reduction of 0.17% in system cost, while increasing storage profit by an average of 10.16% across various system uncertainty scenarios and bidding strategies. These benefits scale up with increased storage capacity withholding and storage capacity.