Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision

Abstract

We introduce a new comprehensive metric of predictable loss (PL) for liquidity providers in constant function automated market makers and derive an optimal liquidity provision strategy. PL compares the value of the LP's holdings in the liquidity pool (assuming no fee revenue) with that of a self-financing portfolio that replicates the LP's holdings and invests in a risk-free account. We provide closed-form formulae for PL, and show that the losses stem from two sources: the convexity cost, which depends on liquidity taking activity and the convexity of the pool's trading function; the opportunity cost, which is due to locking the LP's assets in the pool. For LPs in constant product market makers with concentrated liquidity, we derive a closed-form strategy that dynamically adjusts the range around the exchange rate as a function of market trend, volatility, and liquidity taking activity in the pool. We prove that the profitability of liquidity provision depends on the tradeoff between PL and fee income. Finally, we use Uniswap v3 data to show that LPs have traded at a significant loss, and to show that the out-of-sample performance of our strategy is considerably superior to the historical performance of LPs in the pool we consider.

Keywords: Decentralised Finance, Automated Market Making, Smart Contracts, Concentrated Liquidity, Algorithmic Trading, Market Making, Stochastic Optimal Control, Predictable Loss, Impermanent Loss