Uniswap v3 Concentrated Liquidity & Impermanent Loss Math Guide

1. From Infinite Curves to Bounded Virtual Reserves

In classical constant-product automated market makers like Uniswap v2, liquidity is uniformly allocated along the hyperbolic curve x · y = k across the entire price spectrum from 0 to ∞. As a consequence, only a tiny fraction of the total deposited assets is actively traded at any given moment, resulting in severe capital inefficiency and extreme slippage for large orders.

Uniswap v3 fundamentally reimagines this curve by allowing liquidity providers (LPs) to concentrate their funds within custom price intervals [p_a, p_b]. To preserve the computational elegance of the constant-product invariant while bounding real token reserves, Uniswap v3 introduces virtual reserves:

(x + L / sqrt(p_b)) * (y + L * sqrt(p_a)) = L^2

Where:
L     = Virtual liquidity measure (sqrt(k))
p_a   = Lower price boundary of position
p_b   = Upper price boundary of position
x, y  = Real token balances deposited by the provider
        

When the current market price $P$ falls precisely on the lower bound $p_a$, the pool holds 100% of token $X$ and 0 of token $Y$. Conversely, when $P$ reaches or exceeds the upper bound $p_b$, the pool holds 100% of token $Y$ and 0 of token $X$.

2. Analytical Derivation of Real Reserve Balances

Given a total virtual liquidity parameter $L$ and active price $P = y / x$, the real reserves $x$ and $y$ held by the position are governed by piecewise continuous equations depending on where $P$ sits relative to the boundaries:

Case 1: P <= p_a (Price collapsed below lower boundary)
x = L * (1 / sqrt(p_a) - 1 / sqrt(p_b))
y = 0

Case 2: p_a < P < p_b (Price active within concentrated interval)
x = L * (1 / sqrt(P) - 1 / sqrt(p_b))
y = L * (sqrt(P) - sqrt(p_a))

Case 3: P >= p_b (Price rallied above upper boundary)
x = 0
y = L * (sqrt(p_b) - sqrt(p_a))
          

The total market valuation $V(P)$ of the position, evaluated in units of asset $Y$ (e.g., USD when pairing ETH/USDC), is given by $V(P) = x \cdot P + y$. Substituting the active case yields:

V(P) = L * (2 * sqrt(P) - sqrt(p_a) - P / sqrt(p_b))
        

3. Exact Concentrated Impermanent Loss Equation

Impermanent loss measures the difference in total portfolio valuation between providing liquidity in an AMM versus passively holding (HODL) the exact initial asset quantities deposited at inception $P_0$.

Suppose an LP deposits capital at price $P_0 \in (p_a, p_b)$, establishing initial balances $x_0$ and $y_0$. If the price subsequently shifts to $P_1$, the valuation of the passive HODL strategy is:

V_HODL(P_1) = x_0 * P_1 + y_0
            = L * ((1 / sqrt(P_0) - 1 / sqrt(p_b)) * P_1 + (sqrt(P_0) - sqrt(p_a)))
        

Therefore, the exact percentage Impermanent Loss (IL_v3) is expressed analytically as:

IL_v3(P_1) = ( V_pool(P_1) - V_HODL(P_1) ) / V_HODL(P_1)

Where V_pool(P_1) evaluates according to the 3 piecewise price cases.
        

4. Capital Efficiency Multiplier Formulations

The tremendous appeal of Uniswap v3 stems from its ability to simulate vast pools of capital using modest deposits. The capital efficiency factor compares the liquidity L_v3 achieved inside [p_a, p_b] with the capital required to achieve identical depth across [0, ∞) in Uniswap v2:

Capital Efficiency Multiplier (Centered Geometric Mean):
Multiplier = 1 / (1 - (p_a / p_b)^(1/4))

Or approximate simplified formula:
Multiplier ~ 1 / (1 - sqrt(p_a / p_b))
        

For instance, if an LP sets a price range of \$1,800 to \$2,200 on ETH around a \$2,000 spot:

• p_a / p_b = 1800 / 2200 = 0.818
• √(0.818) = 0.9045
• Efficiency Multiplier ≈ 1 / (1 - 0.9045) = 10.47×

This means \$10,000 deposited into this concentrated range earns the same trading fee volume as \$104,700 deposited in a Uniswap v2 pool, assuming price stays within the boundary!

5. Quantitative Options Equivalence: The Short Strangle / Put Spread

In modern quantitative finance, a concentrated LP position is formally isomorphic to a continuous short option strategy:

• Short Put Component: As the asset depreciates toward $p_a$, the LP automatically buys the asset, ending up 100% long at $p_a$.
• Covered Call Component: As the asset appreciates toward $p_b$, the LP automatically sells into the quoting asset (cash), capping upside at $p_b$.
• Option Premium Equivalent: Continuous protocol trading fees collected by the pool correspond precisely to the theta decay received by an options seller.

By understanding this equivalence, sophisticated market makers hedge their AMM inventory by dynamically shorting futures contracts or buying out-of-the-money puts to maintain delta neutrality.