This AMM Loss Matches an Option’s Time Decay

If you provide liquidity in a trading pool, one hidden cost keeps nibbling at your returns as prices move against you. This paper shows that cost, called loss-versus-rebalancing or LVR, can be described with the language of options, the contracts traders use to bet on future price moves. The authors model a constant-function automated market maker position as a portfolio of exotic options called perpetual American continuous-installment options, and they show that the model reproduces an AMM position’s delta over an infinite time horizon, including the ability to withdraw liquidity. Their main result is striking: the AMM’s LVR is analytically identical to the continuous funding fees, or time value decay (theta), earned by the at-the-money option in the replication. They also derive a special case whose position profile and price boundaries suffer approximately constant LVR, with only a bounded residual error, over an arbitrarily long forward window. The framework also shows how to calibrate the constant volatility parameter from implied volatility data and estimate both calibration error and LVR error, giving liquidity providers a way to choose position settings with more predictable adverse-selection costs.

If you add money to a trading pool, price moves can quietly eat your return. An AMM, short for automated market maker, is a trading pool that sets its own price rule. That loss is called loss-versus-rebalancing, or LVR. It is the bill you pay when the pool trades against you. If you have ever watched a fee gain shrink after a price swing, the feeling is familiar. This work turns that hidden drain into something you can price. The surprise is sharp. The AMM loss matches the time decay of a certain option, called a continuous-installment option. That matters because option decay has a market price. It also works over an infinite time line. It even keeps the choice to pull liquidity out later. That makes the loss feel less like luck.

Why a pool loss can be priced like time decay

The central result links two things that seem far apart. One is LVR, the adverse-selection cost, the loss from trading when prices move first. The other is theta, the time value decay that traders watch in options. The model shows that these are the same quantity for the at-the-money CI option inside the replica. At-the-money means the option sits near the current price. CI options earn or lose value through continuous funding fees, which arrive bit by bit over time. The model also builds a special case. That case gives a delta profile. A delta profile is the way a position reacts as price moves. It also gives price bounds that suffer almost constant LVR. The error stays bounded. So the loss can stay close to flat for an arbitrarily long forward window.

How the option picture is built

The model treats a CFAMM, a pool that follows one fixed price rule, as a basket of options. It uses a perpetual American continuous-installment option as the key piece. Perpetual means no end date. American means the holder can act early. Continuous-installment means the cost arrives in a stream, not one lump sum. The option sits at the money, near the current price. It tracks the AMM's delta, or price sensitivity, at each moment. Delta means how much a position moves when price moves. Because the AMM can last forever, the model keeps that horizon in view. It also keeps the choice to withdraw liquidity in view. That is what makes the match between pool loss and option decay work.

What this means for liquidity providers

For liquidity providers, the gain is practical. The model turns a hidden cost into a cost you can price. It lets a provider choose a liquidity range and a forward window with a target level of LVR. The goal is not zero loss. The goal is loss that stays about the same. The framework also uses implied volatility, the market's own guess of future wiggle size, to set the constant volatility input. It then estimates the error from that fit and the leftover LVR gap. In plain terms, the tool gives a way to plan around adverse-selection costs. It helps users avoid surprise each time the price jumps.

A long view on future loss

The surprise is still the spine. A pool loss can be read like option decay. That makes long-horizon planning possible for a CFAMM position. A provider can pick price bounds, a time window, and a target LVR shape before adding funds. The most concrete payoff is simple. LVR stops looking like an opaque tax on every trade. It becomes a cost that the option model can estimate from implied-volatility data. The next hard check is how well that estimate holds when the implied-volatility curve shifts. If it stays close, the model gives liquidity providers a way to price future pain before they step in.