The Water-Filling Rule

When a cryptocurrency futures exchange faces losses exceeding its insurance fund, it must reduce positions and distribute losses among solvent participants. This auto-deleveraging mechanism is deployed rarely but matters enormously — it determines who bears losses when the system breaks.

arXiv:2603.15963 formulates auto-deleveraging as an optimization problem minimizing the exchange’s risk from future shortfalls. The optimal single-asset policy has a clean form: minimize the maximum leverage among participants, reducing the most leveraged positions first.

The solution is a water-filling algorithm. Imagine leverage as the height of water in columns of different widths. The optimal policy pours water out until all columns reach the same level — the most leveraged accounts are reduced first, and the process stops when risk is equalized. This is not a heuristic. It is the provably optimal solution to the risk minimization problem.

The design resists wash-trading and Sybil attacks because splitting an account into two accounts does not change the water-filling outcome — each fragment gets the same treatment the whole would have received. The solution is also distribution-free: it does not depend on assumptions about the distribution of future price movements.

In multi-asset cross-margin settings, the problem remains separable using asset-level shadow prices. When risk is driven by a dominant factor, the optimal policy takes the same water-filling form but in a factor-adjusted notion of leverage. Hedged portfolios are treated less aggressively than gross leverage alone would suggest — the optimization distinguishes between leverage that creates risk and leverage that manages it.

The elegance comes from the fact that fairness and optimality coincide: the policy that minimizes systemic risk is also the one that treats participants most equitably.