  The `Black Thursday' crisis in cryptocurrency markets demonstrated
deleveraging risks in over-collateralized non-custodial stablecoins. We develop
a stochastic model that helps explain deleveraging crises in these
over-collateralized systems. 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
regimes that are interpreted as stable and unstable for the stablecoin. We
prove bounds on quadratic variation and the probability of large deviations in
the stable domain and we demonstrate distinctly greater price variance in the
unstable domain. We identify a deflationary deleveraging spiral by means of a
submartingale. These deleveraging spirals, which resemble short squeezes, lead
to faster collateral drawdown (and potential shortfalls) and are accompanied by
higher price variance, as experienced on Black Thursday. We conclude by
discussing non-custodial ways in which the issues raised in this paper can be
mitigated.
