While Stability Lasts: A Stochastic Model of Stablecoins

The `Black Thursday' crisis in cryptocurrency markets demonstrated deleveraging risks in over-collateralized lending and stablecoins. We develop a stochastic model of over-collateralized stablecoins that helps explain such crises. In our model, the stablecoin supply is decided by speculators who optimize the profitability of a leveraged position while incorporating the forward-looking cost of collateral liquidations, which involves the endogenous price of the stablecoin. We formally characterize stable and unstable domains for the stablecoin. We prove bounds on the probabilities of large deviations and quadratic variation in the stable domain and distinctly greater price variance in the unstable domain. The unstable domain can be triggered by large deviations, collapsed expectations, and liquidity problems from deleveraging. We formally characterize a deflationary deleveraging spiral as a submartingale that can cause such liquidity problems in a crisis. We also demonstrate `perfect' stability results in idealized settings and discuss mechanisms which could bring realistic settings closer to the idealized stable settings.