> ## Documentation Index
> Fetch the complete documentation index at: https://inflexion.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.
# 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](/protocol/pricing-stack) — [MaxIL](/math/maxil-derivation) → [fairRate / FairPremium](/math/fair-value-phi-sum) → [the load stack](/math/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](/math/il-formula), the [MaxIL convexity proof](/math/maxil-derivation), the [Φ-sum](/math/fair-value-phi-sum), the [load stack](/math/load-stack), and the [no-bad-debt math](/math/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](/concepts/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](/concepts/maxil) is the maximum in-range IL, attained at a range boundary (the [convexity proof](/math/maxil-derivation)):
$$
\mathrm{MaxIL} \;=\; \max\!\big(\mathrm{IL}(P_a),\; \mathrm{IL}(P_b)\big)
$$
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](/math/maxil-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](/math/fair-value-phi-sum) moves between them. This is precisely why MaxIL is the [unit of risk](/concepts/maxil) that makes positions fungible to an underwriter.
## Step 2 — fairRate and FairPremium (three durations)
[fairRate](/math/fair-value-phi-sum) 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](/protocol/oracles); 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](/math/sigma-ref).
## Step 3 — the load stack → premium
The published premium is FairPremium scaled by the [load stack](/math/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](/security/invariants).
**Both rails price off this same FairPremium.** Path A computes the load stack on-chain; Path B (a [market maker](/protocol/path-b-mm)) 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](/protocol/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:
Inside the range the convex IL curve sits **below** its boundary endpoints (the [convexity proof](/math/maxil-derivation)), 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.
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.
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](/math/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](/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](/protocol/routing) end to end.
The signature-free [cvAMM](/protocol/path-a-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.
`createSwapRouted` for swap #2 priced **both rails off the same FairPremium** and chose the [market maker](/protocol/path-b-mm) 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](/reference/deployments) for the full table.
## Where to go next
The rigorous version of Step 1 — why the maximum in-range IL is always at a boundary.
The exact closed form behind Step 2's fairRate — a finite lognormal-moment sum, no Monte Carlo.
baseLoad + the two skews + the I10 clamp — the inventory layer of Step 3.
The three-line proof that the cap binding in scenario C can never become bad debt.
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