CHAPTER 02 / 8 · Understand the mechanics
Watch your assets change.
Predict the asset you will hold above, inside and below a range.
A concentrated-liquidity position describes prices where you are willing to exchange assets. Its token amounts change as swaps move the pool price through that interval.
Jump to the interactive experiment ↓Follow one trade through the curve
Start with 100 ETH and 270,000 USDC, so k = 27,000,000. Ignore fees. A trader buys 10 ETH, leaving 90 ETH. To preserve the product, USDC inventory must become 300,000. The trader therefore pays 30,000 USDC: an average execution price of 3,000 per ETH.
After the trade, the next infinitesimal unit is priced at 300,000 / 90 = 3,333.33. Starting spot (2,700), average fill (3,000) and ending spot (3,333.33) answer different questions. In the experiment, change the amount first, then change the LP depth while keeping that amount fixed.
- x: ETH inventory; y: USDC inventory; P: USDC per ETH.
- q: ETH bought out of the pool, with 0 < q < x.
AssumptionsFull-range constant-product model, no fees; k is fixed during a swap.
TEST IT HERE · CHANGE ONE ASSUMPTION
- Reserve multiplier
- 1.00 ×
- ETH inventory before swap
- 100.00 ETH
- USDC inventory before swap
- 270,000.00 USDC
- Invariant relative to original
- 1.00 ×
Original: 100 ETH and 270,000 USDC. Proportional scaling preserves the starting spot of 2,700 USDC/ETH.
A→B changes liquidity, keeping the reserve ratio and price fixed. B→C is a swap along the selected invariant curve (gold segment). New k is a different hyperbola, not a rigid translation. This models proportional full-range funding, not arbitrary single-token deposits; in concentrated liquidity only overlapping ranges add active depth.
Compare marginal and average execution prices
| Trader pays | 30,000.00 USDC |
|---|---|
| Trader receives | 10.00000 ETH |
| Fee separated from curve reserves | 0.00 USDC |
| Difference versus external reference, before other costs | -3,000.00 USDC |
Each scenario starts from the selected proportional liquidity state; trades do not accumulate. Moving only the external reference changes the comparison, not the curve. This simplified input fee stays outside curve reserves; it is not exact v2 or ADPP accounting. The reference assumes unlimited external depth and no gas or hedge costs, so the difference is not executable arbitrage profit.
LP deposits change the curve, not just the point
Now double both reserves before trading: 200 ETH and 540,000 USDC. The price is still 2,700 but k is four times larger. Buying the same 10 ETH now costs about 28,421.05 USDC, averaging 2,842.11. The deeper pool absorbs the trade with less price movement. Proportional withdrawals have the opposite effect.
This is a move to a different hyperbola, not a rigid translation. It assumes proportional provision to a full-range model. In a concentrated pool, adding liquidity changes depth where the new position is active; an out-of-range deposit does not deepen trading at the current price.
- x: ETH inventory; y: USDC inventory; P: USDC per ETH.
- c: reserve multiplier. 1.5 adds 50%; 0.5 removes half.
AssumptionsProportional entry/exit in the full-range model; neither an unbalanced deposit nor a concentrated range formula.
What changes when you choose a range?
Instead of offering inventory at every price, you choose where to exchange. Throughout this course P means USDC per ETH. For a standard position with lower bound a and upper bound b, principal is all ETH at or below a, a mixture inside, and all USDC at or above b. Accrued fees are separate.
Inside the range, swaps increase ETH inventory as pool price falls and reduce it as price rises. The equations below specify those quantities. They assume fixed liquidity, no fees and a pool price that actually reaches P; changing an external reference price alone is not a swap.
- x: ETH inventory; y: USDC inventory; P: USDC per ETH.
- a, b: lower and upper prices; L: fixed liquidity scale, not USDC value.
AssumptionsInside a < P < b, principal only. No fees, tick rounding or changes in L.
GO DEEPERRead an LP position like an options book.
Start from the reserve equations
Fix liquidity L and bounds a < b, with P in USDC per ETH. Inside the range, the equations below determine both token amounts. Mark principal as V = Px + y. Below a, evaluate ETH inventory at P = a and keep USDC at zero; above b, evaluate USDC inventory at P = b and keep ETH at zero. These are principal-only formulas with no fees, changes to L or tick rounding. They describe a standard curve aligned to the evaluation price, not a pool blocked by a hook from reaching it.
- x: ETH inventory; y: USDC inventory; P: USDC per ETH.
- a, b: lower and upper prices; L: fixed liquidity scale, not USDC value.
AssumptionsInside a < P < b, principal only. No fees, tick rounding or changes in L.
Turn an intention into an inventory path
At spot 3,000, a 2,400–2,800 buy range starts entirely in USDC. If swaps traverse it downward, it progressively buys ETH. Complete traversal spends USDC at an average of √(2,400 × 2,800), about 2,592.30 per ETH before fees. It does not buy the entire amount at the lower endpoint. No traversal means no completed purchase.
To sell ETH gradually, place a range above spot and start with ETH. A range surrounding spot provides both directions and generally needs both assets; their starting value weights depend on the endpoints and are not automatically 50/50.
TRY IT YOURSELF
Make the mechanics visible.
In the mini lab, move the price from 3,000 to 2,500 and back. Compare the tokens, not just the total value. The full Liquidity Lab adds path playback and the one-way policy illustration.
At / above upper endpoint · principal is USDC
Value mix at simulated price · ETH 0.0% / USDC 100.0%
10,000 USDC initial value at 3,000 USDC/ETH. Uses the existing Lab math. Assumes swaps follow the simulated price; excludes fees, gas, MEV and tick rounding. Changing width creates a new funded scenario, not a live rebalance.
Why an accumulation can reverse
Ordinary liquidity is reversible. After the buy range acquires ETH, a rebound through it can sell that ETH for USDC again. A fixed-range, fee-free position returning to the same pool price returns to the same principal quantities, provided liquidity has not changed. Actual fees and intervention make the total outcome different.
This is why a range is not automatically a permanent acquisition order. If you want to retain acquired ETH, define when to withdraw or agree a compatible directional pool policy. Such a policy still cannot guarantee counterparties or completion by a date.
CHECK YOUR UNDERSTANDING
What does this mean for you?
Is your goal to keep acquired ETH, reduce ETH exposure, or repeatedly quote both sides?
CONNECT IT TO AMM MARKETS
From understanding to a real decision.
Use the AMM Markets Liquidity Lab to test that intention before choosing a live range. Dedicated ADPP policies can be discussed with our team when ordinary two-way liquidity does not fit your objective.
Open the full Liquidity Lab ↗Educational simulation, no real funds.References and scope
Course edition: October 2026. Distinguish protocol principles from product instructions; actual market parameters come from the live interface and on-chain state.