CHAPTER 03 / 8 · Understand the mechanics
Choose concentration deliberately.
Distinguish deposited capital, active liquidity and execution depth.
Two positions with the same capital can provide very different depth. Concentrated liquidity lets you decide where that capital participates, at the cost of a more specific inventory and monitoring commitment.
Jump to the interactive experiment ↓The question is not just “how much capital?”
Compare two positions funded with 10,000 USDC of value at ETH = 3,000: a narrow 2,850–3,150 range and a wide 2,400–3,600 range. Both have the same budget. The narrow position commits it to exchange over fewer prices, so more inventory changes hands per small price move while it is active.
A wider range keeps serving trades across a larger movement, but supplies less local depth per invested dollar. Choose between them by specifying where you actually want to buy and sell, not by maximizing the tallest bar.
Why concentration creates depth
Liquidity L is a scale parameter, not a dollar balance. Inside a fixed range, the inventory equations are proportional to L. At a given price, narrowing the endpoints reduces the capital required for one unit of L, so the same capital buys more L.
For a small price increase, net USDC entering the curve is approximately L / (2√P) times the price change. More L means more input is needed for the same move. Across initialized ticks, add the liquidity active in each segment and calculate the trade segment by segment; one local formula is not a quote across every range.
- δy: net USDC entering the curve; δP: a small increase in USDC/ETH price.
- P: starting price; L: total active liquidity in this segment.
AssumptionsLocal approximation at constant L, before fees. Cross-tick trades need segment-by-segment calculation.
TRY IT YOURSELF
Make the mechanics visible.
Compare equal 10,000 USDC starting values below. Switch between narrow and wide around 3,000, then move price to 3,300. Which position is still active? Inspect the full depth chart later in Markets.
Inside range · both assets
Value mix at simulated price · ETH 48.8% / USDC 51.2%
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.
More depth also means faster inventory conversion
At equal capital, the narrow position’s ETH inventory falls faster when price rises and grows faster when it falls, as long as both positions remain active. In options language, its local negative Gamma has larger magnitude. Unhedged LPs also retain positive Delta: they still bear directional ETH risk.
At 3,300, the narrow example is entirely USDC and no longer earns swap fees there; the wide example still holds both assets and participates. Below the lower endpoint a position is entirely ETH. Zero active liquidity or zero Gamma outside the range does not mean zero portfolio risk.
GO DEEPERRead an LP position like an options book.
Delta is the inventory you actually hold
Differentiate principal value with respect to price: Delta is the ETH inventory, and Gamma is the rate at which that inventory changes. The equations and units are set out below.
A price increase reduces ETH inventory; a decrease increases it. Below range Delta stays at the all-ETH quantity, while above range it is zero. Outside range Gamma is zero, but all-ETH price risk can remain substantial. V and Delta are continuous at the boundaries; Gamma jumps and its two-sided derivative is not defined exactly there.
- V: principal value in USDC; Δ: ETH; Γ: ETH per (USDC/ETH).
AssumptionsFixed L and a < P < b. Outside, Γ = 0; exactly at a boundary the two-sided Gamma is undefined. Not a complete hedged-return model.
Concentration magnifies local exposure per dollar
For fixed entry capital, a narrower range needs larger L, so active |Gamma| increases. Compare positions at the same entry price and budget, not at equal L. For a small in-range move δP, δV ≈ ΔδP + ½Γ(δP)². The second term is negative. It explains curvature relative to a tangent, not the entire P&L. Across a boundary or a large jump, use the exact piecewise values; a local Taylor approximation can be badly wrong.
Covered calls: a shape analogy, not an identity
As price rises through the range, ETH converts to USDC and upside eventually caps in USDC terms. This resembles a covered call at expiry. But a vanilla covered call has a strike, expiry and an upfront option premium; LP conversion is distributed across prices, has no built-in expiry and earns uncertain transaction-dependent fees. Its inventory can reverse on a retrace. Option time value, implied volatility and settlement terms cannot be inferred from the LP shape alone.
Short straddles and delta hedging
A delta-hedged LP is locally short convexity, which may feel familiar to an options seller. An unhedged LP is generally long Delta as well: it is not a delta-neutral short straddle. A plain straddle has a kinked terminal payoff and different tail, collateral and expiry mechanics. Under a fixed-curve, zero-rate illustration, V(P) can be represented as a linear inventory term plus a continuum of short calls weighted by −V″(K). This is a static value-function representation, not a claim that one listed option replicates LP cash flows or that an AMM fee is an option premium.
Separate capital, depth and a trade quote
TVL marks the principal across positions, including capital far from spot. Active liquidity describes the segment currently serving trades. An execution-depth curve asks how much input moves the pool to another price through the available segments. These are related measurements, not interchangeable units.
AMM Markets shows pool-derived asset value, liquidity distribution, execution-price illustrations and policy bounds separately. A tall relative depth chart alone cannot tell you a 20,000 USDC swap will execute. For a real trade, fees, price bounds, token conditions and current state must also pass the Router’s checks.
Choose a range you can manage through a path
Start with three scenarios: price stays inside, trends above, or falls below and remains there. For each, write the asset you will hold, whether swaps can still earn fees, and what would justify an intervention. Estimate gas and any swaps needed to move the range.
A narrow position can earn a larger active fee share but spend less time active and incur more management cost. A wide position is not automatically safer in every sense: it commits inventory to exchange over more prices. The decision must combine your intended inventory, plausible flow and net costs.
CHECK YOUR UNDERSTANDING
What does this mean for you?
Would you accept holding only ETH after a fall? How often could you review the position?
CONNECT IT TO AMM MARKETS
From understanding to a real decision.
In Markets, compare asset quantities and TVL with Pool liquidity depth and the execution-price illustration. Read the loaded window and snapshot time. An ADPP price bound is a separate overlay; it does not erase liquidity outside it.
Inspect market depth ↗Opens a live service. Exploring is separate from submitting a transaction.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.