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Pricing and Hedging for Liquidity Provision in Constant Function Market Making

Authors: Jimmy Risk, Shen-Ning Tung, Tai-Ho Wang Year: 2026 (March 2026) arXiv: 2603.01344 Categories: q-fin.MF

Plain-language abstract

In DeFi, Constant Function Market Makers (CFMMs) like Uniswap and Balancer replace the traditional limit order book with a deterministic bonding curve \(f(x, y) = K\) relating token reserves to prices. A Liquidity Provider (LP) deposits tokens into a smart contract and effectively sells them to the market according to this rule — so an LP position is really a derivative, not a passive deposit. This paper reframes CFMM analysis from the usual (token reserves) coordinates into a (price, intrinsic liquidity) coordinate system that is dimensionally consistent across bonding curves. Under this coordinate change, reserves and value functions become linear in the liquidity profile, which in turn lets the authors characterise Impermanent Loss as a weighted strip of vanilla options via the Carr–Madan spanning formula. Empirical validation on Uniswap v3 ETH/USDC pools vs Deribit option implied-vol surfaces confirms a crypto-consistent volatility smile.

Key contributions

  1. Dimensionally consistent "intrinsic liquidity" — the standard \(K\) in bonding curves like \(\sqrt{xy} = K\) (CPMM) or \(x^\alpha y^{1-\alpha} = K\) (G3M) carries different physical dimensions across protocols, so you cannot compare liquidity across them numerically. The authors define a local intrinsic liquidity \(\ell(x, y)\) that always has dimension \(\sqrt{\text{ETH} \times \text{USDC}}\), regardless of the functional form of \(f\).
  2. Canonical parametrization theorem — reserves \((x, y)\) can be recovered from the pair (spot price \(p\), local intrinsic liquidity profile \(\ell(p)\)) via integral representations, independent of the bonding-curve parametrisation.
  3. Linear structure — mark-to-market value \(V_L(p)\) is linear in the liquidity profile \(L(q)\), which simplifies arbitrage-free pricing, delta hedging, and systematic risk management.
  4. IL as weighted options strip — via Carr–Madan, Impermanent Loss can be decomposed into a weighted strip of vanilla calls/puts with weights set by \(L(q)\). This gives a fine-structure implied volatility for liquidity profiles.
  5. Path-dependent IL via last-passage time — provides granular risk analysis of when IL is actually realised.
  6. Empirical consistency: Uniswap v3 ETH/USDC pool data vs Deribit option markets shows a volatility smile consistent with crypto-asset dynamics.

Method summary

Coordinate change: reserves → (price, liquidity)

For a smooth bonding curve \(f(x, y) = K\), the local intrinsic liquidity at reserve state \((x, y)\) is defined via partial derivatives of \(f\) so that the result always carries dimension \(\sqrt{\text{ETH}\times\text{USDC}}\) — matching CPMM's \(K\) regardless of the underlying form.

  • Locality: \(\ell\) depends on the current reserve point (it is state-dependent, unlike \(K\)).
  • Invariance: \(\ell\) is intrinsic to the geometry of the level curve; unchanged under reparametrisations like \(xy = K^2\) vs \(\sqrt{xy} = K\).

The canonical parametrization (Theorem 2.2)

Given smooth, strictly increasing, convex \(f\), the reserves at price \(p\) are:

\[x(p) = \int_p^\infty \frac{L(q)}{q} \, dq, \qquad y(p) = \int_0^p q \cdot L(q) \, dq \cdot \text{(with appropriate scaling)}\]

where \(L(q) = \ell(q)/q\) is the liquidity profile. This expresses the CFMM in a coordinate system that is protocol-agnostic.

LP value as covered-call strip

Mark-to-market value with token \(Y\) as numéraire:

\[V_L(p) = \int_0^\infty \min\{p, q\} \cdot L(q) \, dq\]

Since \(\min\{p, q\} = p - (p - q)^+\) is a covered call payoff, \(V_L\) is a weighted strip of covered calls indexed by \(L(q)\). This is the bridge to derivatives pricing.

Impermanent Loss via Carr–Madan

Using Carr–Madan's spanning formula, IL decomposes into a weighted combination of vanilla options, yielding a risk-neutral decomposition amenable to delta hedging with standard option instruments.

Empirical validation

  • Uniswap v3 ETH/USDC pool liquidity profiles.
  • Deribit ETH options for comparison implied-vol smile.
  • Fitted fine-structure IV from LP positions matches crypto-asset volatility smile properties.

Main results

  • LP positions in Uniswap v2 (CPMM) and Uniswap v3 (concentrated liquidity) unify under the same price–liquidity coordinate system.
  • Impermanent Loss is exactly hedgeable via a static option strip whose weights are determined by the current liquidity profile.
  • The implied-vol structure extracted from LP fee streams is consistent with Deribit-observed crypto vol smile.
  • The framework subsumes / recovers existing partial results (Fukasawa et al. variance-swap IL hedge, Angeris–Evans–Chitra CFMM payoff space, Loss-Versus-Rebalancing as continuous-installment option, etc.).

Limitations

  • Geometry assumes smooth, convex, increasing \(f\) — some newer AMM invariants (proactive market makers with discontinuities) are out of scope.
  • Continuous-time setting: discrete fee events, gas costs, and MEV extraction are abstracted away.
  • Single-pool analysis: cross-pool arbitrage, multi-asset baskets (Balancer n-token pools beyond G3M), and routing across DEX aggregators not modelled.
  • No informed-flow / adverse-selection model: LPs in reality lose systematically to arbitrageurs (Loss-Versus-Rebalancing); this paper prices LP positions risk-neutrally but does not model the structural drag of trading against informed flow in equilibrium.
  • Empirical scope: ETH/USDC on Uniswap v3 only; alt-L2 pools and stable-stable pools have different dynamics.

Connections

  • This is market making in the AMM paradigm — a parallel world to traditional LOB market making. See market-making for the combined picture.
  • Crypto-native setting complements the centralised-exchange crypto work in explainable-crypto-microstructure, which studies Binance Futures perpetuals (order-book venue). The two together span the main venues where LPs operate.
  • Adverse selection is the conceptual link to traditional microstructure: Loss-Versus-Rebalancing is the AMM analogue of Glosten–Milgrom adverse selection — LPs lose to informed arbitrageurs who rebalance pools when prices move. See adverse-selection.
  • Carr–Madan spanning sits in the broader derivatives literature; this paper's main engineering value is the coordinate change that makes that classical tool apply cleanly to LP positions.