Worked example end-to-end

One 50k LP position, ±10% range, three durations and three terminal prices — computing MaxIL, fairRate, FairPremium, load, premium, and the settlement payout for each, then the real on-chain 2026-06-05 lifecycle numbers for both rails.

This page takes the whole stack out for a drive. We start from a single concrete LP position, walk the pricing stack — MaxIL → fairRate / FairPremium → the load stack → premium — for three durations, then settle the same position at three terminal prices. Every intermediate number is the verified output of the Monte-Carlo reference that the on-chain Stylus oracle matches to machine precision, and the page closes on the real on-chain create→settle numbers logged on 2026-06-05 for both rails.

The single mental model to carry through it:

$$\text{premium} \;=\; \underbrace{\text{fairRate}}_{\text{stochastics: how risky}} \;\cdot\; \underbrace{\mathrm{MaxIL}}_{\text{geometry: how much capital}} \;\cdot\; \underbrace{(1 + \text{load})}_{\text{inventory: how scarce}}$$

This is the applied companion to the conceptual math pages. If a step is unfamiliar, jump to its home page: the IL formula, the MaxIL convexity proof, the Φ-sum, the load stack, and the no-bad-debt math.

The position

A Uniswap v3 LP holds an ETH/USDC position worth $V_0 = 50{,}000$ USDC at entry, with a geometric-symmetric ±10% range around the entry price. To make every figure concrete we pin the entry price at $P_0 = 3{,}000$ USDC/ETH; the dollar results scale with $V_0$ and are otherwise $P_0$-independent in the fair-rate sense.

Parameter	Value
Entry price $P_0$	$3{,}000$ USDC/ETH
Range width $w$	$\pm 10\%$ (geometric)
Lower bound $P_a = P_0/(1+w)$	$2{,}727.27$
Upper bound $P_b = P_0\,(1+w)$	$3{,}300.00$
Entry value $V_0$	$50{,}000$ USDC
In-range requirement	$P_a \le P_0 \le P_b$ ✓ (enforced at creation)

The in-range gate $P_a \le P_0 \le P_b$ is what makes this an in-range convexity hedge rather than insurance — an out-of-range entry is rejected at createSwap because there is no in-range IL left to hedge.

Solving $V_{\text{hold}}(P_0) = 50{,}000$ for $L$ at $P_0 = 3{,}000$ gives the entry basket $a_0 = 8.3\overline{3}$ ETH and $a_1 = 25{,}000$ USDC — a 50/50 value split at entry, exactly as expected for a centered range. Everything below is derived from this $(P_a, P_b, P_0, L)$ tuple.

Step 1 — MaxIL (pure geometry, one number for all three durations)

MaxIL is the maximum in-range IL, attained at a range boundary (the convexity proof):

$$\mathrm{MaxIL} \;=\; \max\!\big(\mathrm{IL}(P_a),\; \mathrm{IL}(P_b)\big)$$

Computing both boundaries for this position:

Boundary	$\mathrm{IL}$ (USDC)	as % of $V_0$
$\mathrm{IL}(P_a = 2{,}727.27)$	$1{,}163.44$	$2.33\%$
$\mathrm{IL}(P_b = 3{,}300.00)$	$\mathbf{1{,}279.78}$	$\mathbf{2.56\%}$
MaxIL $= \max(\cdot)$	$\mathbf{1{,}279.78 \approx 1{,}280}$	$\mathbf{2.56\%}$

Two facts to read off this table:

1. The upper boundary wins. $\mathrm{IL}(P_b) > \mathrm{IL}(P_a)$ even though the range is ratio-symmetric — the structural upper-boundary asymmetry (affine $V_{\text{hold}}$ growing without bound vs. flattening $V_{\text{lp}}$, see the derivation). $\mathrm{MaxIL} = \mathrm{IL}(P_b)$.
2. MaxIL/V0 = 2.56% matches the reference magnitude for a ±10% geometric range exactly.

MaxIL is frozen here, once, and it is identical for all three durations. It is pure geometry — no $\sigma$, no $T$ enters it. The 7d, 30d, and 90d quotes below are all backed by the same $1{,}280$ USDC of collateral; only fairRate moves between them. This is precisely why MaxIL is the unit of risk that makes positions fungible to an underwriter.

Step 2 — fairRate and FairPremium (three durations)

fairRate is the dimensionless fraction of MaxIL the capped claim is worth under the risk-neutral measure — an exact closed-form Φ-sum, an S-curve in $\sigma^2 T$:

$$\text{fairRate} \;=\; \frac{E_Q[\min(\mathrm{IL}, \mathrm{MaxIL})]}{\mathrm{MaxIL}} \in [0, 1], \qquad \text{FairPremium} \;=\; \text{fairRate}\cdot \mathrm{MaxIL}$$

Evaluating at $\sigma_{\text{ref}} = 60\%$ for the ±10% geometry across the three durations:

Duration	fairRate	FairPremium $= \text{fairRate}\cdot 1{,}280$	as % of $V_0$
7d	$44.9\%$	$574.72 \approx \mathbf{574}$ USDC	$1.15\%$
30d	$70.8\%$	$906.24 \approx \mathbf{906}$ USDC	$1.81\%$
90d	$82.9\%$	$1{,}061.12 \approx \mathbf{1{,}061}$ USDC	$2.12\%$

Read this as the S-curve walking up $\sigma^2 T$: over 7d the ±10% band is rarely touched, so the claim is worth only 44.9% of MaxIL (the most convexity value, the cheapest fraction); over 90d the band is exited most of the time, so the claim is worth 82.9% of MaxIL — you are pre-paying a near-certain loss. Over 90d the band is touched ~75% of the time vs. ~24% over 7d, and the S-curve does all the work.

$\sigma_{\text{ref}}$ is the only stochastic input. It is the conservative max(σ_short, σ_long, floor) from the VolOracle; here we use the calibrated illustrative $60\%$. On the live deploy the oracle initialized to its $0.50$ floor (fairRate ≈ 0.847 at floor vol). See σ_ref.

Step 3 — the load stack → premium

The published premium is FairPremium scaled by the load stack, hard-capped by invariant I10:

$$\text{premium} \;=\; \text{FairPremium}\cdot\big(1 + \underbrace{\text{baseLoad} + \text{util\_skew} + \text{dispersion\_skew}}_{\text{clamped to maxLoad}}\big)$$

For this walkthrough take a clean +15% total load (skews = 0, a normal-regime baseLoad example — the live calibrated baseLoad bands are 20/30/50% by σ regime). The premium the cvAMM publishes for the same position at each duration:

Duration	fairRate	FairPremium	cvAMM premium (@ +15% load)	as % of $V_0$
7d	$44.9\%$	$574$ USDC	$660.93 \approx \mathbf{661}$ USDC	$1.32\%$
30d	$70.8\%$	$906$ USDC	$1{,}041.78 \approx \mathbf{1{,}042}$ USDC	$2.08\%$
90d	$82.9\%$	$1{,}061$ USDC	$1{,}220.29 \approx \mathbf{1{,}221}$ USDC	$2.44\%$

Load sensitivity on the 30d cell (the load is the only thing the pool’s inventory state can move): at +5% load the 30d premium is $906 \times 1.05 = \mathbf{952}$ USDC; at +30% it is $906 \times 1.30 = \mathbf{1{,}178}$ USDC. The ceiling is $\text{FairPremium}\cdot(1+\text{maxLoad})$, enforced by the I10 clamp.

Both rails price off this same FairPremium. Path A computes the load stack on-chain; Path B (a market maker) carries a single loadBps and the contract derives premium = FairPremium·(1 + loadBps/1e4), requiring loadBps ≤ maxLoadBps (I10 on Path B). createSwapRouted gives the LP the cheaper of {pool, MM} and falls back to the pool if the MM quote is unusable — see routing.

Step 4 — settlement at three terminal prices

The LP buys the 30d quote: premium $1{,}042$ USDC, collateral locked $1{,}280$ USDC (= MaxIL). At expiry the protocol pins the settlement price $P_T$ on-chain from the Chainlink round-at-expiry and pays the LP $\min(\text{realized\_IL}, \mathrm{MaxIL})$ from the underwriter’s collateral:

$$\text{realized\_IL} \;=\; \max\!\big(0,\; V_{\text{hold}}(P_T) - V_{\text{lp}}(P_T)\big), \qquad \text{payout} \;=\; \min(\text{realized\_IL}, \mathrm{MaxIL})$$

We settle the same position at three representative terminal prices: one inside the range, one at the upper boundary, and one beyond the range where the cap binds hard.

Scenario	$P_T$	$V_{\text{hold}}$	$V_{\text{lp}}$	realized IL	payout $=\min(\mathrm{IL},\mathrm{MaxIL})$
A — interior ($-4\%$)	$2{,}880.00$	$49{,}000.00$	$48{,}780.71$	$219.29$	$\mathbf{219.29}$ (uncapped)
B — at boundary ($+10\%$, $P_T = P_b$)	$3{,}300.00$	$52{,}500.00$	$51{,}220.22$	$1{,}279.78$	$\mathbf{1{,}279.78}$ (= MaxIL exactly)
C — beyond range ($+25\%$)	$3{,}750.00$	$56{,}250.00$	$51{,}220.22$	$5{,}029.78$	$\mathbf{1{,}279.78}$ (capped — cap binds)

Three things this table shows, each load-bearing:

1. A — interior: the payoff is exactly the realized IL

   Inside the range the convex IL curve sits below its boundary endpoints (the convexity proof), so realized IL ($219.29$) is well under MaxIL and the LP is paid it in full, uncapped. Net of the $1{,}042$ premium the LP’s hedge cost this scenario is $1{,}042 - 219.29 = 822.71$ USDC — the price of the volatility protection in a calm outcome.

2. B — at the boundary: payout equals MaxIL by construction

   At $P_T = P_b$ the realized IL equals MaxIL to the cent ($1{,}279.78$) — this is the definition of MaxIL, the maximum in-range IL attained at a boundary. The payout is the full collateral; the underwriter keeps $\text{collateral} - \text{payout} = 0$ residual on this swap.

3. C — beyond range: the cap is load-bearing, not a defect

   At $+25\%$ the LP has fully rotated into USDC ($V_{\text{lp}}$ is flat above $P_b$), while $V_{\text{hold}}$ keeps growing — realized IL balloons to $5{,}029.78$, but the payout is capped at $1{,}279.78$ = MaxIL. The LP eats the $3{,}750.00$ of uncovered loss. That uncovered slice is directional loss — foregone spot upside the LP would have had holding ETH — not the impermanent loss the product set out to hedge. Capping at MaxIL is exactly what keeps the product fully collateralized and trustless.

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.

In every scenario above, $\text{payout} \le \text{collateral} = \mathrm{MaxIL} = 1{,}280$ — invariants I1 (no bad debt) and I2 (cap correctness), structurally and for any price path. The cap binding in scenario C is the visible form of the no-bad-debt guarantee: the protocol never owes more than it pre-locked. See the no-bad-debt math.

Limit → Why → Mitigation. Scenario C looks like the LP is “under-protected” by $3{,}750$. Why: beyond the range the loss is directional (held-asset opportunity cost), structurally distinct from in-range IL and unbounded — there is no finite collateral that could insure it. Mitigation: the cap is disclosed up front on every LP surface (payoff diagram + “in-range convexity hedge” label), so the truncated tail never surprises an LP, and the LP keeps the choice of a wider range (larger MaxIL, more coverage) at a higher premium. See what this is not.

The whole stack on one position, one screen

Collecting Steps 1–4 for the 30d quote, end to end:

$$\underbrace{\mathrm{MaxIL} = 1{,}280}_{\text{geometry}} \;\xrightarrow{\;\times\,\text{fairRate}=0.708\;}\; \underbrace{\text{FairPremium} = 906}_{\text{stochastics}} \;\xrightarrow{\;\times\,(1+0.15)\;}\; \underbrace{\text{premium} = 1{,}042}_{\text{inventory}}$$ $$\text{settle at } P_T = 3{,}750:\quad \text{realized\_IL} = 5{,}029.78 \;\xrightarrow{\;\min(\cdot,\,\mathrm{MaxIL})\;}\; \text{payout} = 1{,}280 \;\le\; \text{collateral} = 1{,}280$$

The only number that changed between the 7d / 30d / 90d quotes was fairRate ($0.449 \to 0.708 \to 0.829$); MaxIL, and therefore the collateral and the cap, was constant throughout.

The real on-chain lifecycle — 2026-06-05, both rails

The illustrative numbers above are reproduced by a live create→settle run on the fresh stack, on a short (300s) fee-500 market, settled with no time-warp at the Chainlink round bracketing each expiry. Two swaps ran — one on each rail — demonstrating the floor-of-liquidity / ceiling-of-price routing end to end.

Path A — cvAMM pool

The signature-free cvAMM underwrote swap #1 directly; settlement paid from the ConvexityVault.

Field	Value
swapId	1 (LP tokenId 3220)
$V_0$	$270,531.28
MaxIL (collateral locked)	$1,669.24
premium	$9.70 (0.58% of MaxIL)
realized IL at settlement	$148.64
payout	$148.64 — paid from ConvexityVault

The premium being a tiny 0.58% of MaxIL is the convexity hedge in the wild: a wide/short position has a low fairRate, so the protection is cheap relative to the worst-case cap. Realized IL ($148.64) landed well inside the cap, so the full uncapped amount was paid and the pool kept $1,669.24 − $148.64 = $1,520.60 residual collateral.

Path B — MM won the route

createSwapRouted for swap #2 priced both rails off the same FairPremium and chose the market maker because its firm signed quote strictly beat the pool. The MM’s own collateral was locked and paid the settlement from the UnderwriterVault.

Field	Value
swapId	2 (LP tokenId 3221)
cvAMM price (premiumA)	$13.80
MM price (premiumB, won)	$8.93 (loadBps = 1000)
chosen path	B (MM) — strictly beat the pool
MaxIL (MM collateral locked)	$3,215.65
realized IL at settlement	$245.66
payout	$245.66 — paid from MM collateral

The MM won by $4.87 ($8.93 vs $13.80). Because the route is “cheaper-of, ties to Path A,” and only the executed rail mutates nonce / capacity / lock, the unchosen pool quote left zero trace (invariant I7). The LP got the better price; the MM’s collateral — not the pool’s — bore the $245.66 payout. This is the ceiling-of-price rail working exactly as designed.

Depositor capital is NOT guaranteed: the junior tranche is first-loss; the senior tranche is structurally protected from underwriting loss only, not systemic tail.

Underwriting capital — whether supplied to the pooled cvAMM vault (Path A) or posted directly by a market maker (Path B) — is at risk:

• Junior tranche — first-loss. Junior depositors absorb underwriting losses first and buffer the senior tranche.
• Senior tranche — protected from underwriting loss, exposed to the systemic tail. The senior tranche is structurally shielded from ordinary underwriting loss while junior capital buffers, but it still takes the systemic tail once the junior buffer is exhausted.

No-bad-debt is a guarantee to the LP being paid (see the FULL-mode payout clause), not a guarantee that depositor or MM capital is preserved.

In both live swaps, $\text{payout} \le \text{collateral} = \mathrm{MaxIL}$ held (I1/I2), and the rail that priced the swap is the rail whose collateral paid it — Path A from the pooled, dual-tranche ConvexityVault; Path B from the single MM’s UnderwriterVault. The settlement core is identical regardless of which rail opened the swap.

The live addresses for every contract referenced here are in the deployment registry. See deployments for the full table.

Where to go next

MaxIL & the convexity proof

The rigorous version of Step 1 — why the maximum in-range IL is always at a boundary.

FairValue — the Φ-sum

The exact closed form behind Step 2’s fairRate — a finite lognormal-moment sum, no Monte Carlo.

The load stack

baseLoad + the two skews + the I10 clamp — the inventory layer of Step 3.

No-bad-debt math

The three-line proof that the cap binding in scenario C can never become bad debt.