Liquidity Pool Position

An automated market maker is a pool holding two assets. Anyone can trade against it, and the price it offers comes from a formula rather than from an order book. Traders pay a fee on every swap.

You can put money into that pool. You deposit both assets, and in return you earn a share of the fees.

Here is the part that surprises people. The formula keeps the pool balanced by value. When one asset rises, the pool sells it — automatically, to the traders taking the other side. When one falls, the pool buys more of it.

So you end up holding more of whatever went down and less of whatever went up. Against simply holding the two assets in your wallet, that is a loss. It has a misleading name: impermanent loss. It is only impermanent if the prices come back.

What you are short

The position is short volatility of the ratio between the two assets. Fees are the premium you collect; divergence is the payoff you owe. That is the same shape as writing a straddle, and it should be read the same way.

• Two assets that move together — a stablecoin pair, say — diverge little, so the fees dominate and the position is close to a lending position.
• Two assets that move independently — the fees have to be large to cover the divergence, and frequently are not.
• A one-way move is the worst case: the pool has sold the whole rally on the way up.

Concentrated liquidity changes the size, not the sign

Newer designs let a provider choose a price range rather than supplying across all prices. Fees per unit of capital rise sharply inside the range and fall to zero outside it. The exposure is more levered in both directions, and the position now needs managing rather than leaving.

The constant-product result

For a pool holding \(x\) and \(y\) with \(xy = k\), the value of the position against simply holding the initial basket, as a function of the price ratio \(p\) relative to entry:

The inequality is strict except at \(p = 1\). The divergence cost is therefore certain in sign and depends only on how far the ratio has travelled, not on the path — which is what distinguishes it from a genuine trading loss.

The break-even condition

with \(\phi\) the fee rate and \(\nu\) turnover per unit of pooled value. The position is profitable when realised volume is high relative to realised divergence — which is the same statement as a market maker's, because that is what the position is.

The risks that are not the formula

Contract risk, oracle and governance risk, and the fact that the position is only as durable as the code holding it. These are unrelated to the payoff above and have historically produced the total losses; the divergence cost has produced the disappointing ones.

The formulas above are standard textbook formulations, simplified for teaching. They explain the mechanism — they are not a valuation tool, and they will not reproduce a dealer’s price.