On-chain published fair value

FairPremium = fairRate · MaxIL, where fairRate is the exact closed-form Φ-sum S-curve in σ²·T — published on-chain by the Stylus FairValueOracle, never reimplemented off-chain.

Inflexion is the first protocol to price impermanent-loss risk on-chain and publish that price as a public good. Pillar 1 of the three pillars is the FairValueOracle: a production contract that computes, live per quote, the fair value of a capped in-range IL claim and exposes it to both rails of the market.

This page is the conceptual tour. It explains what the fair value is, why it factorizes the way it does, and why it is an exact closed form rather than a fitted surface, a Monte-Carlo simulation, or a lookup table. The full term-by-term derivation lives in the Φ-sum derivation.

The headline identity

Everything reduces to one multiplication:

$$\textbf{FairPremium} \;=\; \textbf{fairRate} \;\cdot\; \textbf{MaxIL}$$

Two factors, two completely separate jobs:

MaxIL — pure geometry

The maximum in-range IL of the specific position, computable at creation from (Pa, Pb, L, P0). Frozen at creation, identical across all three durations, and carries none of the vol/time dependence.

fairRate — pure stochastics

The fraction of MaxIL the claim is worth under the risk-neutral measure. An S-curve in σ²·T. It carries all of the vol/time dependence; geometry enters it only through the normalized range edges.

A one-line mental model that runs through the whole pricing stack: MaxIL is the how much capital (geometry); fairRate is the how risky (stochastics); the load stack is the how scarce (inventory state). They multiply.

One position, three prices, one collateral. Because fairRate carries all the time dependence and MaxIL carries none, a single Uniswap position prices to three different FairPremium values (7 / 30 / 90 days) — all backed by the same MaxIL. Only fairRate moves with duration. See the worked example below.

fairRate: the S-curve in σ²·T

fairRate is defined as the risk-neutral expectation of the capped payoff, normalized by the cap:

$$\textbf{fairRate} \;=\; \frac{\mathbb{E}_{\mathbb{Q}}\!\left[\min(\mathrm{IL},\, \mathrm{MaxIL})\right]}{\mathrm{MaxIL}} \;\in\; [0,\,1]$$

It is dimensionless — the probability-weighted fraction of the cap you expect to be paid. As a function of total variance σ²·T, it traces a clean S-curve between two asymptotes:

• As σ²·T → 0 (calm and/or short): fairRate → 0. Price barely moves, almost never leaves the range, the cap is irrelevant — the claim is near-worthless.
• As σ²·T → ∞ (violent and/or long): fairRate → 1. Price almost surely exits the range and the claim almost surely pays the full MaxIL — you are effectively pre-paying a near-certain loss.

The entire pricing surface lives in the S-shape between those asymptotes, and it is governed jointly by total variance and by how wide / close-to-edge the range geometry is.

Reference points (σ = 60%, risk-neutral)

These are the verified outputs of the reference IL ground truth, reproduced by cvamm.fair_rate. Read each row as the S-curve walking up σ²·T with duration:

Range width	7d	30d	90d
±5%	69.5%	84.8%	91.3%
±10%	44.9%	70.8%	82.9%
±20%	18.2%	47.3%	67.4%

A wide / short position (±20% / 7d) is worth only 18% of MaxIL — the cheapest fraction, and the most convexity value for the LP. A tight / long position (±5% / 90d) is worth 91% of MaxIL — effectively pre-paying a near-certain loss. The product carries the most convexity value for wide/short positions and the least for tight/long ones.

This is why premium is quoted as a % of MaxIL rather than a % of position value. Because MaxIL is the collateral the underwriter posts, charging X% of MaxIL gives the underwriter the same return-on-collateral regardless of range width — so it is indifferent to width, and depth never fragments. The full argument is on the MaxIL page.

Why it is an EXACT closed form

This is the most important claim behind pricing IL on-chain. The answer: fairRate has no calibrated coefficients. It is not fit to data, not simulated, and not interpolated from a table. The only stochastic input is σ_ref (the volatility reference).

fairRate is NOT a fitted surface, a Monte-Carlo estimate, or a lookup table — it is a finite, exact sum of normal-CDF terms (Φ), evaluated live per quote.

The mechanism, in one paragraph. The capped v3 payoff min(IL, MaxIL) is piecewise in the terminal price P_T — three arms, split by the range edges and the two cap-crossing prices:

• below Pa — IL is linear and decreasing in P_T (the position has rotated fully into token0), capped flat at MaxIL once it crosses the lower cap price;
• inside [Pa, Pb] — IL is the convex arm c₁·P − 2L·√P + c₀, and is never capped (a convex function on the range sits below its endpoint chord, which is ≤ MaxIL);
• above Pb — IL is linear and increasing in P_T (rotated fully into token1, while a hold grows), capped flat at MaxIL above the upper cap price.

Under risk-neutral GBM, P_T is lognormal, so each arm integrated against the density is a standard lognormal interval moment expressible in the normal CDF Φ. The building block is one closed expression:

$$\mathbb{E}\!\left[ P_T^{\,p}\cdot \mathbf{1}\{K_1 < P_T < K_2\}\right] = P_0^{\,p}\,e^{\frac12 p(p-1)\sigma^2 T}\,\bigl(\Phi(d(K_1)) - \Phi(d(K_2))\bigr)$$ $$\text{with}\quad d(K) = \frac{\ln(P_0/K) + (p - \tfrac12)\,\sigma^2 T}{\sigma\sqrt{T}}$$

The arms use only the moments p ∈ {0, ½, 1}, and the integration limits are the finite set K ∈ {0, Pa, Pb, PcapL, PcapR, +∞}. Summing the arms gives:

$$\textbf{FairPremium} = \mathbb{E}_{\mathbb{Q}}\!\left[\min(\mathrm{IL}, \mathrm{MaxIL})\right] \;=\; \text{a finite } \Phi\text{-sum (}\approx\text{6–10 terms, Black–Scholes class)}$$

No Monte Carlo. No lookup table. No fitted coefficients. Evaluated live, per quote.

The Φ-sum is L-independent. It runs in normalized units P0 = 1, L = 1, so it depends only on a = Pa/P0, b = Pb/P0, σ_ref, and T — not on the position’s liquidity. This L-independence in the fair-rate sense is exactly what makes positions fungible to an underwriter within a market, and why an MM quote is per-market, never per-NFT.

Verified exact against the settle-path math

The closed form is checked equal to the reference IL implementation — which is itself verified equal to dense quadrature and to Monte Carlo — so the pricer can never drift from the settlement math:

• Closed form vs reference quadrature / MC: max_abs_err ≈ 5.1×10⁻¹¹. The heavy calibration run restates this as ≡ the reference IL implementation to 3.3×10⁻⁹ — both far inside any tolerance.
• Stylus Φ-sum vs a 50-digit mpmath reference: the production Rust matches the high-precision fixtures to ≤ 1×10⁻¹² on fairRate.

The Stylus FairValueOracle is production

A hard design rule: the Φ-sum is never reimplemented off-chain. The SDK reads the on-chain FairPremium and never approximates it. The on-chain pricer is the Stylus FairValueOracle — a Rust/WASM contract whose integer fixed-point erf is machine-precise precisely because Stylus bans WASM floating point at activation.

Production — Stylus FairValueOracle

Rust/WASM, machine-precise Φ-sum (≤ 1×10⁻¹² on fairRate vs the 50-digit reference). The single source of fair value for both rails — the router pokes once and prices Path A and Path B off the same FairPremium. Address in the deployment registry.

CI cross-check — Solidity oracle

The Solidity FairValueOracle uses an Abramowitz–Stegun Φ (~1.5×10⁻⁷, ÷MaxIL-amplified at the tight tier). It is a revm-testable CI cross-check, not a second production oracle.

At swap creation the protocol calls FairValueOracle.fairPremium(oracleToken, a, b, durationSeconds, maxIL), which returns (premium, fairRateWad, sigmaRefWad). The same read feeds both the cvAMM load stack (Path A) and the MM loadBps derivation (Path B), so the routing compares two prices built from one fair value.

Both rails build their premium on top of this published number — Path A as FairPremium · (1 + load stack) clamped by invariant I10, Path B as FairPremium · (1 + loadBps) with loadBps ≤ maxLoadBps. The fair value is the floor both rails are honest about. See the pricing stack.

The only residual approximation

The Φ-sum is exact given the model. There is exactly one modeling assumption, and it is stated plainly:

Limit: the closed form assumes risk-neutral geometric Brownian motion with μ = 0, r = 0 (forward = P0). No on-chain formula removes this assumption — it is the price of having an exact, coefficient-free expression.

Why it’s acceptable: GBM with zero drift is the standard, model-light risk-neutral baseline for a short-dated claim, and the residual it leaves is forward-looking-vol information, not a structural error.

Mitigation — two layers:

1. A conservative σ_ref. The vol input is max(σ_short, σ_long, floor) with the floor deliberately set above calm ETH vol, so the pool is net-conservative; over the heavy run, realized payout/premium came in at 0.654. See σ_ref.
2. MM alpha by design. The residual forward-vol premium is deliberately left for Path-B market makers, who price off implied / forward vol and correct the pool’s structural backward-looking bias. That blindness is the incentive that keeps MMs in the two-sided market — a feature, not a bug.

It is worth being precise about what σ_ref is and is not load-bearing for. A wrong or too-low σ_ref makes the pool under-charge load, so premium income stops covering payouts and NAV compresses — junior absorbs it first, senior in the tail. It is not load-bearing for the FULL no-bad-debt invariant: in FULL, collateral = MaxIL ≥ payout regardless of what σ was, so a vol-oracle fault can cost depositors money but can never create LP bad debt. These are two separate guarantees and are never merged.

LPs are always paid with no bad debt in FULL mode — qualified: capped payoff + solvent USDC + oracle/settlement liveness.

The no-bad-debt guarantee is exact only under the full clause:

• FULL collateralization — the underwriter has posted exactly MaxIL per swap, locked for the life of the position.
• Capped payoff — the payout is min(realized_IL, MaxIL). The cap is load-bearing: it is what makes payout ≤ collateral hold by construction.
• Solvent USDC — the collateral numeraire (dUSDC) holds its peg and remains redeemable.
• Oracle / settlement liveness — the price oracle stays live and settle() runs within the protocol’s settlement window.
• No rehypothecation breach — locked collateral stays instantly liquid; it is never routed to utilization-gated venues.

Stated without these qualifiers, the guarantee is not correct.

Worked example: one position, three prices

A 50,000 USDC position, ±10% geometric range, σ_ref = 60%, baseLoad = +15%, skews = 0. MaxIL = 1,280 USDC (2.56% of V0) — identical for all three durations. Only fairRate moves.

Duration	fairRate	FairPremium	cvAMM publishes (@ +15% load)
7d	44.9%	574 USDC	661 USDC (1.32% of V0)
30d	70.8%	906 USDC	1,042 USDC (2.08% of V0)
90d	82.9%	1,061 USDC	1,221 USDC (2.44% of V0)

The collateral the underwriter posts (MaxIL = 1,280 USDC) is the same across the table; the S-curve does all the work. Over 90 days the ±10% band is touched ~75% of the time versus ~24% over 7 days — which is the fairRate column, made concrete.

Live on-chain

The pricer priced live swaps on the Arbitrum Sepolia stack (2026-06-05):

Path A — cvAMM (swap #1)

V0          = $270,531.28
MaxIL       = $1,669.24
premium     = $9.70   (0.58% of MaxIL — a wide/short position, low fairRate)
realized IL = $148.64
payout      = $148.64  (paid from the ConvexityVault)

Both rails priced off the same published FairPremium; createSwapRouted gave the LP the cheaper of the two. The full address registry and lifecycle record live in the deployment reference.

Theory anchors — not the pricer

An important distinction: two academic results are the anchors for why the claim is priceable and hedgeable. They are not the on-chain pricer — the exact Φ-sum is. Do not conflate them.

Lipton, Lucic & Sepp (2025) — static replication

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. This is the theory anchor for why the claim is priceable and hedgeable. It is not the formula Inflexion evaluates on-chain. See theory anchors.

Milionis, Moallemi & Roughgarden (2022) — LVR

The AMM’s adverse-selection cost (loss-versus-rebalancing) has a closed form proportional to instantaneous variance — equivalently the theta of the replicating short-option position. A v3 LP is structurally short gamma (concave V_lp), and IL is the realization of that short-gamma / LVR cost. Inflexion sells the LP the long-convexity leg that offsets it, capped at MaxIL. See the in-range convexity hedge.

Where this sits in the stack

MaxIL — the cap and unit of risk

The geometry factor: why the cap is load-bearing and why it makes positions fungible.

The Φ-sum derivation [DEEP]

The full term-by-term derivation of the closed form deferred from this page.

σ_ref — the volatility oracle

The only stochastic input: max(σ_short, σ_long, floor) and why the max is mandatory.

The pricing stack

How both rails build their premium on top of the published FairPremium.