6.11.3 Concentrated Liquidity & Uniswap v3

Basic Prerequisites: 02 · The Constant-Product AMM.

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1. Intuition & Practical Objective

Uniswap v3’s concentrated liquidity solves the constant-product AMM’s core inefficiency: a full-range LP’s capital is spread over all prices, but only prices near the current one are ever traded. v3 lets an LP concentrate capital inside a chosen price range $[P_a,P_b]$ (a set of adjacent ticks), so the same dollars quote a deeper book in-range - at the cost of the position converting to a single token the moment price exits the range.

The practical objective: understand the token-amount formulas, the LP value function, and the sharp structural fact that a concentrated position is a bounded short-gamma / short-strangle payoff - the divergence loss is truncated by the range bounds, which is both its appeal (capped downside relative to a levered full-range position of equal depth) and its trap (when price exits, you are 100% in the loser’s token).

The three ideas:

1. Concentration is leverage. Within the range the position behaves like a larger full-range CPMM of virtual reserves $x_v,y_v$ (with $L=\sqrt{x_v{y}_v}$ the “liquidity”). Concentrating into a narrow range multiplies per-dollar depth - and multiplies per-dollar short gamma and divergence loss proportionally.
2. The range bounds the optionality. Outside $[P_a,P_b]$ the position is fully in one token and its value is linear in price - gamma is zero. The short-straddle of 02 is now a strangle truncated at the strikes $P_a,P_b$.
3. This is the options-replication story made operational. The v3 position’s concave value function in-range and linear (flat-gamma) value out-of-range is the Clark (2020) short-option portfolio with the tails cut off by the LP’s chosen strikes.

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2. Mathematical Ground Truth & Derivations

2.1 Virtual reserves and the liquidity parameter

A v3 position inside $[P_a,P_b]$ acts like a constant-product pool with virtual reserves $x_v,y_v$ ($x_v{y}_v=L^2$) offset so that the real, spendable balances hit zero exactly at the bounds. The real token amounts held at price $P$ are (Uniswap v3 Core, §6):

$$x(P)=L\left(\frac{1}{\sqrt{P}}-\frac{1}{\sqrt{P_b}}\right),y(P)=L\left(\sqrt{P}-\sqrt{P_a}\right),$$

valid for $P\in [P_a,P_b]$; at $P\le P_a$ the position is all $X$ with $x=L(\frac{1}{\sqrt{P_a}}-\frac{1}{\sqrt{P_b}})$; at $P\ge P_b$ it is all $Y$ with $y=L(\sqrt{P_b}-\sqrt{P_a})$. Here $L$ is fixed when the position is opened. The marginal price is still $p=y_v/x_v$, and the depth at any in-range price scales with $L$.

2.2 LP value function and bounded convexity

The position value in numeraire $Y$ is

$$V(P)=x(P)P+y(P).$$

Differentiating twice in-range gives a negative second derivative (concavity / short gamma); the magnitude of the concavity grows as the range narrows (concentration multiplies gamma per dollar). Crucially, outside the range $V$ is linear in $P$ (position is a single token), so the second difference vanishes - the short optionality is truncated at the strikes $P_a,P_b$:

$$\frac{d^{2}V}{d{P}^2}<0\text{for }P\in (P_a,P_b),\frac{d^{2}V}{d{P}^2}=0\text{for }P\notin [P_a,P_b].$$

This is the exact signature of a short strangle: concave (short) convexity between the strikes, no convexity outside. Compared with a full-range pool funded by the same initial deposit, the concentrated position carries more divergence loss per dollar in-range (more leverage) but the loss is capped - once price exits, no further divergence loss accrues, only directional exposure in the remaining token.

2.3 Divergence loss of a concentrated position

For the position opened at $P_0$ inside $[P_a,P_b]$, the divergence-loss fraction is still “1 minus LP value over hold value,” but now the LP value is the truncated value function above. Because $V$ is linear outside the range, the marginal divergence loss of further adverse movement is zero beyond the bounds - the position has already fully converted. The same closed-form core as 02 applies, but the range cap changes both the worst case (bounded) and the realistic outcome (if the price exits and doesn’t return, the position is stuck in the depreciating token).

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3. Computational Implementation - v3 token formulas and the gamma signature

We compute the v3 token amounts and value across a range of prices, and measure the concavity (numerical second difference) in-range versus out-of-range.

Two facts jump out. First, the position converts exactly at the bounds: at $P\le 0.5$ it is all $X$ (x=70.711, y=0), at $P\ge 2.0$ all $Y$ (y=70.711), and in-range it holds a balanced mix (at $P=1$, $x=y=29.289$). Second, the concavity is negative in-range ($-2.026$ - the position loses value at a convex rate as price swings, i.e. it is short gamma) and exactly zero out-of-range (the position is linear, so no more divergence loss accrues). That is the bounded short-strangle signature of concentrated liquidity, and it is exactly the payoff the Clark (2020) replication predicts.

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4. Failure Modes & First-Principles Breakdowns

1. Range drift is the silent killer. If price exits $[P_a,P_b]$ and keeps going, the LP holds 100% of one token and earns zero fees (no trading happens in-range). “Concentrated liquidity” means concentrated timing risk: you must actively re-position or you end up fully in the loser.
2. Concentration multiplies divergence loss per dollar. A narrow range gives deeper quotes in-range but proportionally more short gamma - and if the range is too tight, even a small adverse move realizes the capped loss while the fees (earned only while price is in-range) were small. The leverage cuts both ways.
3. Active management is now required. v3 LPs must monitor the price and migrate liquidity (paying gas each time). The “passive AMM market maker” of v2 becomes a semi-active gamma trader - the inventory / skewing problem of Avellaneda–Stoikov reappears on-chain, only now with gas costs and no cancel (rebalance) latency.
4. The capped loss is still a real loss. Because the position converts to a single token, the divergence loss is not recovered if price returns: v3 LPs who got fully converted and never rebalanced experienced permanent, not impermanent, losses.

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5. References

• Adams, Zinsmeister, Salem, Keefer & Robinson (2021), Uniswap v3 Core, Uniswap whitepaper
• Clark, Joseph (2020), The Replicating Portfolio of a Constant Product Market, SSRN 3550601
• Capponi & Jia (2022), The Anatomy of a Liquidity Provision in Automated Market Makers, arXiv:2210.07852
• Lehar & Parlour (2023), Decentralized Exchange: The Uniswap Automated Market Maker

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6. Connected Graph Bridges

• Back: 02 · The Constant-Product AMM
• Forward: 04 · MEV & Sandwiching · Index Hub
• Theory: Options Fundamentals · Black–Scholes–Merton · Market Impact & Depth