Unified approach for hedging impermanent loss of liquidity provision

We develop static model-independent and dynamic model-dependent approaches for hedging of the impermanent loss (IL) of liquidity provision (LP) staked at Decentralised Exchanges (DEXes) which employ Uniswap V2 and V3 protocols. We provide detailed definitions and formulas for computing the IL to unify the different definitions occurring in the existing literature. We show that the IL can be seen as a contingent claim with a non-linear pay-off for a fixed maturity date. Thus, we introduce the contingent claim termed the IL protection claim which delivers the negative of IL pay-off at the maturity date. We apply arbitrage-based methods for the valuation and risk management of this claim. First, we develop the static model-independent replication method for the valuation of IL protection claim using traded European vanilla call and put options. We extend and generalise an existing method to show that the IL protection claim can be hedged perfectly with options if there is a liquid options market. Second, we develop the dynamic model-based approach for the valuation and hedging of IL protection claims under a risk-neutral measure. We derive analytic valuation formulas using a wide class of price dynamics for which the characteristic function is available under the risk-neutral measure. As base cases, we derive analytic valuation formulas for the IL protection claim under the Black–Scholes–Merton model and the log-normal stochastic volatility model. We finally discuss the estimation of the risk–reward of LP staking using our results.