Methodology

1 · Economic setting and notation

Consider liquidity provision between a risky asset and a stablecoin, with ETH-stablecoin as the leading example. The stablecoin is the numéraire, and the listed option surface is taken on the risky asset. More generally the same construction applies to any underlying with a sufficiently liquid option surface.

A concentrated-liquidity position is defined over a price interval [S_ℓ, S_u] with 0 < S_ℓ < S_u. Inside the interval the position continuously rebalances between the two assets; outside it, the provider is effectively fully invested in one side of the pair. The object of interest is the loss of the position relative to a passive benchmark holding the same inventory without rebalancing. That loss is path-dependent after the fact, but under option-replication arguments it can be linked to the risk-neutral distribution of future prices — and therefore inferred from option markets.

Corridors

Ranges are taken symmetrically around the forward price F_τ for horizon τ. For a corridor parameter α ∈ (0,1):

R⁻_α = (1 − α)F_τ , R⁺_α = (1 + α)F_τ The index is evaluated across α ∈ {0.10, 0.15, 0.20, …, 0.90}, producing a surface over horizon and range width rather than a single number.

Notation

`S`	Spot price of the risky underlying
τ	Target maturity (for example, 30 days)
`F`_τ	Forward price of the risky underlying at the target maturity
`O`(`K`, τ)	Out-of-the-money option price at strike `K` and maturity τ
`R`⁻_α, `R`⁺_α	Lower and upper corridor boundaries
σ_imp(`K`, τ)	Black-76 implied volatility at strike `K` and maturity τ
SVI	Stochastic Volatility Inspired parameterization

The time subscript is suppressed throughout; all calculations are snapshots at a fixed point in time.

2 · Volatility surface construction

The valuation integrals require option prices at arbitrary strikes and at a fixed horizon, neither of which the market quotes directly. The surface is therefore reconstructed before it is integrated.

SVI total variance

For each snapshot and listed maturity, parameters (a, b, ρ, m, σ) are estimated from market data. In log-moneyness k = log(K/F_τ), total variance and implied volatility are

w(k) = a + b ( ρ(k − m) + √(k − m)² + σ² ) , σ_imp(K, τ) = √w(k)τ Before evaluation, log-moneyness is clipped to ±3·√a + bσ and total variance is capped at 20. Both are numerical safeguards against explosive tail values, not additional modeling assumptions.

Moneyness grid

Each maturity slice is evaluated on a fixed grid of forward moneyness m = K/F_τ, spanning roughly m ∈ [0.3, 2.0], with strikes K_i = F_τ m_i.

Interpolation across maturities

The target horizon rarely coincides with a listed slice. The two bracketing maturities τ_low < τ^★ < τ_high are interpolated in total variance, not in volatility, with the forward interpolated linearly:

w^★(K) = w_low(K) + λ ( w_high(K) − w_low(K) ) , λ = τ^★ − τ_lowτ_high − τ_low The target implied volatility follows as σ^★ = √w^★(K)/τ^★.

Delta filtering

Black-76 forward deltas are computed on the interpolated smile and the support is restricted to a moderate range — 0.01 ≤ Δ_call ≤ 0.99 and −0.99 ≤ Δ_put ≤ −0.01 — so that the pricing integrals are evaluated only where the smile is numerically reliable rather than on deep-tail extrapolation.

Black-76 pricing

d₁ = log(F/K) + ½σ²τ σ√τ , d₂ = d₁ − σ√τ C = DF·(F N(d₁) − K N(d₂)) and P = DF·(K N(−d₂) − F N(−d₁)). The out-of-the-money price is O(K, τ) = P for K < F and C for K ≥ F. In the current implementation the rate is zero, so discounted and undiscounted prices coincide.

3 · Implied impermanent loss

Both expressions below are risk-neutral expectations of the liquidity position's underperformance, written as a weighted strip of out-of-the-money options.

Concentrated liquidity (v3)

IILX(α, τ) = 12τ R⁺_αR⁻_α∫ O(K, τ) K^3/2 ( 2√K − √R⁻_α − K√R⁺_α ) dK Only strikes inside the corridor enter the integral. Note what the bracket in the denominator is: the value of the liquidity position when the price ends at K — so each option is weighted by how much that price level matters to the position.

Full range, constant product (v2)

IILX(v2, τ) = 14τ ∫ O(K, τ)K² dK With no corridor the weight collapses to 1/K² across the whole supported surface — the weighting of a variance swap.

Consistency check

A variance swap has fair value (2/τ)∫O(K)/K²dK, so the full-range expression is exactly one eighth of it: IILX(v2, τ) = σ²_implied/8. That recovers the standard constant-product result — impermanent loss accrues at roughly an eighth of variance — with implied variance in place of a historical estimate. The realized side satisfies the same identity, which is why the realized full-range series is computed as ⅛ of annualized return variance.

4 · Where the value comes from

The dashed line is the weight each strike receives in the integral; the shaded line is its actual contribution once multiplied by the option price traded there. Narrowing α truncates the corridor and removes the tails from the calculation entirely.

Option strip weighting

Illustrative: forward normalized to 1, flat volatility, 30-day horizon. Both curves are scaled to their own maximum so the shapes are comparable.

Corridor α — ±30%

Implied volatility — 60%

Design

5 · Realized impermanent loss and the premium

The realized series applies the same band geometry to spot prices sampled at regular intervals, typically hourly. With S₀ the current spot and S₁ the next observation, the range is re-centered each step: S_ℓ = (1 − α)S₀, S_u = (1 + α)S₀.

Position value

V_P(S₁) = L̃ { S₁( 1/√S_ℓ − 1/√S_u ) S₁ < S_ℓ 2√S₁ − √S_ℓ − S₁/√S_u S_ℓ ≤ S₁ < S_u √S_u − √S_ℓ S₁ ≥ S_u L̃ is a liquidity scaling constant; it cancels in the ratio below.

Buy-and-hold benchmark

V_H(S₀, S₁) = L̃ { S₁( 1/√S_ℓ − 1/√S_u ) S₀ < S_ℓ (S₀ + S₁)/√S₀ − √S_ℓ − S₁/√S_u S_ℓ ≤ S₀ < S_u √S_u − √S_ℓ S₀ ≥ S_u

Aggregation

The one-step loss and its annualized rolling average are

RIL^step = − V_P − V_HV_H , RIL(α, m) = 365A · 1W Σ RIL^step For a target horizon of m days at observation frequency with annualization constant A (A = 24 for hourly data), the rolling window is W = mA, subject to a minimum-data requirement. The full-range case is RIL(v2, m) = ⅛ · 365A · Var(ΔS/S) over the same window.

The premium

The impermanent-loss risk premium is the difference between the two sides, ILRP = IIL − RIL, for matching underlying, horizon and range width. IIL measures the level of exposure; ILRP measures the compensation for bearing it.

6 · Numerical implementation

Integrals are evaluated on a discrete strike grid K₁ < … < K_n with trapezoidal-type weights

ΔK₁ = K₂ − K₁, ΔK_n = K_n − K_n−1, ΔK_i = K_i+1 − K_i−12 On an equally spaced grid this coincides with the standard trapezoidal rule. The corridor grid spans [R⁻_α, R⁺_α]; the full-range grid spans the supported smile.

Workflow

Load market data — SVI parameters, forward, spot and maturity per snapshot.
Select target maturity — exact slice if available, otherwise interpolate between the nearest bracketing slices.
Construct the smile — evaluate SVI on the moneyness grid and convert to implied volatilities.
Filter tails — retain only strikes whose deltas lie in the stable range.
Price options — Black-76 call and put prices on the forward, taking the out-of-the-money side.
Integrate — the corridor formula for each α, plus the full-range benchmark.

7 · References

White paper	IILX — The Derive Implied Impermanent Loss Index. L. Schönleber, A. Papanicolaou, S. Dawson. Download PDF →
Derivations	The Implied Impermanent Loss in Decentralized Liquidity Provision. A. Papanicolaou, L. Schönleber, T. Li. SSRN → Implied Impermanent Loss for Concentrated Liquidity. L. Alberici, A. Papanicolaou, L. Schönleber.
Methods	Gatheral (2004, 2006), SVI parameterization · Black (1976), forward option pricing · Adams et al. (2021), concentrated liquidity · Heimbach et al. (2022), liquidity-provision risk