Read 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.

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.

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.