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.