This paper introduces a market-based framework to study the effects of tail climate risks in the financial sector. In addition to identifying the financial institutions most vulnerable to physical and transition climate risks, our framework explores the potential for these risks to induce contagion effects in the financial sector. Based on the securities of large European financial institutions spanning from 2005 to 2022, we show that, unlike physical risk, transition risk significantly and increasingly influences systemic risk in the financial sector. We also examine the potential levers available to financial institutions and regulators to address climate-related financial risk.