We propose a theoretical framework to understand the price stability mechanism of stablecoins. With the model, we identify three sources of price instability that form a trilemma. The trilemma suggests that any stablecoin design can avoid at most two of all the following risks: (1) downward price instability due to moral hazards of the operating entity, (2) downward price instability when the entity is exposed to external market risk and has a poor financial performance, and (3) upward price instability caused by limited coin supply. When evaluated using our theoretical framework, many existing stablecoins are found to suffer from the trilemma as predicted by our theory. We further conduct a large-scale global survey of 17,550 individuals from 34 countries to see how the general public perceives the risks of the trilemma. The survey finds that heterogeneity exists across countries and that most people perceive the potential price instabilities influenced by moral hazards and financial risks are larger than the upward price instability. Our study uncovers a fundamental principle of the price stability mechanism in stablecoins, identifying a critical choice for stablecoin issuers and justifying regulatory interventions.