This paper introduces a market-based framework to study the effects of tail climate risks in the financial sector. We identify the financial institutions most exposed to physical and transition climate-related shocks, and analyze how these shocks may propagate through the financial network and amplify systemic risk. Using security-level data on large European financial institutions from 2005 to 2022, we show that transition risk, unlike physical risk, significantly and increasingly contributes to systemic risk. We also examine the potential levers available to financial institutions and regulators to address climate-related financial risk.