Theory anchors — LVR & static replication

The two academic results that make IL priceable and hedgeable — Lipton-Lucic-Sepp replication and Milionis-Moallemi-Roughgarden LVR — and why they validate the approach but are NOT the pricer.

Inflexion does not invent a pricing theory. It applies two established results to a v3 LP position and then prices the exact, capped payoff with a closed-form Φ-sum. This page covers the two anchors, the short-gamma chain that ties them to the IL claim, and — emphatically — why they validate the approach but are not the on-chain pricer.

One-line takeaway. The literature tells us an LP is short gamma and that its loss is the realization of that short-gamma cost. That is why the claim is priceable and hedgeable. The price itself comes from the exact Φ-sum — E_Q[min(IL, MaxIL)] evaluated live per quote — not from these papers. Anchors justify; the Φ-sum computes.

The chain in one diagram

The whole intellectual argument is a five-link chain. Each link is either a property of the v3 value function (proven in the IL formula and the MaxIL derivation) or one of the two academic anchors.

1. V_lp is concave in price

   Inside the range, the v3 LP value $V_{lp}(P) = L\,(2\sqrt{P} - \sqrt{P_a} - P/\sqrt{P_b})$ is strictly concave: $d^2 V_{lp}/dP^2 = -\tfrac14 L\,P^{-3/2} < 0$ for all $P>0$. Holding the entry basket, $V_{hold}(P)$, is affine. Concave value vs. an affine benchmark is the source of IL.

2. Concave value ⇒ the LP is short gamma

   Negative second derivative of value with respect to price is the definition of negative gamma. A v3 LP is structurally short gamma — they have sold optionality to the pool.

3. Short gamma ⇒ the LP pays LVR (MMR 2022)

   Milionis-Moallemi-Roughgarden show the AMM’s adverse-selection cost — loss-versus-rebalancing (LVR) — has a closed form proportional to instantaneous variance, equivalently the theta (time-decay) of the replicating short-option position. Short gamma is paid for in continuous theta bleed; that bleed is LVR.

4. LVR realized over the position ⇒ IL owed

   Integrated over the life of the position, that short-gamma / LVR cost is what shows up at settlement as realized impermanent loss, $\text{realized\_IL} = \max(0,\,V_{hold} - V_{lp})$.

5. Inflexion sells the offsetting long-convexity leg (LLS 2025)

   Lipton-Lucic-Sepp show an IL-protection claim is statically replicable by a strip of vanilla options — so it has a model-light fair value and a concrete hedge. Inflexion sells the LP exactly that long-convexity leg, capped at the in-range maximum (MaxIL), and prices it as the risk-neutral expectation of the capped payoff.

The compressed form:

$V_{lp}$ concave ⇒ LP short gamma ⇒ pays LVR ⇒ owes IL; Inflexion sells the LP the long-convexity leg that offsets it, capped at the in-range maximum (MaxIL), and prices it with the exact risk-neutral expectation of that capped payoff (the Φ-sum).

Anchor 1 — Lipton, Lucic & Sepp (2025): static replication

Result. An IL-protection claim on a concentrated-liquidity position is statically replicable by a strip of vanilla options.

Why it matters for Inflexion. Static replication is a strong statement: it means the claim’s payoff can be reproduced, at expiry, by a fixed portfolio of vanilla calls/puts chosen once — no dynamic rebalancing required. Two consequences fall straight out:

Model-light fair value

A statically replicable payoff inherits the option strip’s price. Its fair value does not depend on a bespoke, calibrated model — it is the value of a known portfolio of vanillas. This is the licence to publish a fair value at all.

A concrete hedge exists

Replication is constructive: the same strip an underwriter could buy to neutralize the risk it has written. This is what makes the written risk a hedgeable book, not an actuarial bet — and it underpins the pool hedge lever on the roadmap.

The v3 LP value function makes the replication tangible. Recall the three regimes (see the IL formula):

$$V_{lp}(P) = \begin{cases} L\,(1/\sqrt{P_a} - 1/\sqrt{P_b})\cdot P & P < P_a \quad(\text{linear}) \\[4pt] L\,(2\sqrt{P} - P/\sqrt{P_b} - \sqrt{P_a}) & P_a \le P \le P_b \quad(\text{concave}) \\[4pt] L\,(\sqrt{P_b} - \sqrt{P_a}) & P > P_b \quad(\text{constant}) \end{cases}$$

The concave middle arm is precisely a short-option payoff profile (capped on the upside, sloping into both tails), and the kinks at $P_a$ and $P_b$ are where the replicating strip’s strikes sit. That this concave value can be written as a portfolio of vanillas is the LLS statement applied to v3.

Anchor 2 — Milionis, Moallemi & Roughgarden (2022): LVR

Result. In Automated Market Making and Loss-Versus-Rebalancing, the AMM’s adverse-selection cost — LVR — has a closed form proportional to instantaneous variance, equivalently the theta of the replicating short-option position.

Why it matters for Inflexion. LVR is the closed-form anchor for the cost of short-gamma exposure. Schematically, the instantaneous LVR rate scales with $\sigma^2$ and the position’s curvature:

$$\text{LVR rate} \;\propto\; \tfrac12\,\sigma^2\,P^2\,\big|\,\partial^2 V_{lp}/\partial P^2\,\big| \;=\; \Theta_{\text{short option}}$$

This is the precise sense in which a v3 LP “bleeds”: holding a concave position through volatility costs theta continuously, and that theta is the LVR. Realized IL is the cumulative realization of this cost over the position’s life. The variance-proportionality is also the bridge to the protocol’s only stochastic input — the same $\sigma$ that drives LVR is the $\sigma_{ref}$ the σ_ref oracle feeds into the price.

The two anchors are complementary, not redundant. MMR gives the cost of being short gamma (the theta you bleed). LLS gives the cure — a static strip that buys the offsetting long-convexity. Inflexion sells the LP that cure. One paper tells you the disease is priceable; the other tells you it is hedgeable.

Why “capped” is where Inflexion departs from the textbook

The anchors describe the uncapped IL claim. Inflexion does not sell that. It sells $\min(\text{realized\_IL}, \text{MaxIL})$ — the in-range maximum, frozen at creation. The reason is structural, not cosmetic: beyond the range the LP has fully rotated into one asset, so $V_{lp}$ is constant while $V_{hold}$ keeps growing linearly, making absolute IL unbounded above $P_b$.

$$\text{covered\_payoff} = \min(\text{realized\_IL},\, \text{MaxIL}), \qquad \text{MaxIL} = \max\big(IL(P_a),\, IL(P_b)\big)$$

The cap truncates the replicating strip at the range boundaries. Everything inside the range is the convexity hedge the anchors describe; everything beyond it is directional loss (foregone spot upside), not the impermanent loss the LP set out to hedge. The cap is what makes the product fully collateralized and trustless — it is the no-bad-debt guarantee, not a defect. This is why Inflexion is an in-range convexity hedge, not “IL insurance.”

The anchors validate the approach — they are NOT the pricer

This is the load-bearing distinction on this page. State it cleanly.

The papers justify that the claim is priceable. They do not set the price. The on-chain price is computed by the exact closed-form Φ-sum — FairPremium = E_Q[min(IL, MaxIL)] — evaluated live per quote in the Stylus FairValueOracle. No coefficient in that formula is fitted to LLS or MMR; the anchors are intellectual provenance, not calibration inputs.

• Limit: the anchors are continuous-time, asymptotic, uncapped results — they describe the family of the claim, not the discrete capped payoff Inflexion actually settles.
• Why: Inflexion settles $\min(\text{realized\_IL}, \text{MaxIL})$ in integer Q64.96 math at a single expiry, against a price the oracle reports — not a continuous-time integral of an uncapped claim.
• Mitigation: the protocol prices the exact capped payoff directly. The capped v3 payoff is piecewise-linear/convex in the terminal price, so under risk-neutral GBM each arm integrates to a standard lognormal interval moment in the normal CDF $\Phi$. The result is a finite Φ-sum (≈6–10 terms, Black–Scholes class), verified ≡ the reference IL implementation to max_abs_err ≈ 5.1e-11 — no Monte Carlo, no lookup table, no fitted coefficients. See the Φ-sum.

The cleanest way to hold the two roles apart:

Role	What it is	Where it lives
Anchor (justifies)	LLS static replication + MMR LVR-as-theta — the claim is priceable & hedgeable	Academic literature
Pricer (computes)	Exact closed-form Φ-sum, E_Q[min(IL, MaxIL)], evaluated live per quote	Stylus FairValueOracle

The only residual modeling assumption

The Φ-sum is exact given the model. The single modeling assumption is risk-neutral GBM with $\mu = 0,\ r = 0$ (measure: risk_neutral_gbm_mu0_r0). The anchors do not remove this; nothing on-chain does. It is covered two ways:

1. The conservative σ_ref

The vol oracle publishes $\sigma_{ref} = \max(\sigma_{short}, \sigma_{long}, \text{floor})$ with a floor of 0.50 sitting above observed calm ETH vol (~0.45), so the pool is net-conservative to depositors. A wrong/too-low $\sigma$ can compress NAV but can never create LP bad debt in FULL — see σ_ref and no-bad-debt math.

2. Forward-vol premium is left as MM alpha

The residual forward-looking-vol premium — the gap between backward-looking realized vol and implied — is deliberately left as market-maker alpha. Forward-vol MMs price off implied vol and correct the pool’s structural backward-looking bias. See Path B — MM.

What this buys the protocol

Provenance, not hand-waving

“IL is priceable” is not an Inflexion claim — it is the LLS + MMR result applied to v3. The protocol stands on published 2022/2025 results.

A real hedge target

Static replication names the offsetting option strip, which is the pool-hedge lever on the roadmap (h≈0.60 tightens the depositor tail).

An honest boundary

The anchors are uncapped & asymptotic; the pricer is exact & capped. Keeping the two roles separate is the rigor — the price is never “an approximation of a paper.”

Related reading

The Φ-sum (the actual pricer)

The exact closed-form fair value the anchors point toward but do not compute.

IL formula

Where $V_{lp}$ concavity (short gamma) and $V_{hold}$ affinity come from.

MaxIL — the cap & unit of risk

Why the in-range maximum is load-bearing and where the strip is truncated.

In-range convexity hedge

The product framing the anchors validate — convexity hedge, not IL insurance.