Understanding Pseudo-Delta Neutral Liquidity Provision

1. Background

In classical option theory, a delta-neutral portfolio is one whose value does not change for infinitesimal moves in the underlying asset price (see [5]). For instance, by shorting a call option and holding an appropriate quantity of the underlying asset, a trader can offset first-order price risk (delta), leaving only higher-order sensitivities (gamma, vega). The essential idea is to construct a portfolio that earns time value (the option premium) while remaining hedged against small price fluctuations.

A similar logic applies in concentrated liquidity market makers. When an LP provides liquidity in a narrow price band, the token composition of their position changes as the pool price evolves (see [3]). This changing composition creates an implicit delta. Below the entry price, the position accumulates more of the risky asset, e.g., SOL, and, above the entry price, it accumulates more of the stablecoin, e.g., USDC. By borrowing SOL or USDC (or both) against collateral, an LP can offset their initial token exposure at the entry price, effectively constructing a pseudo-delta-neutral position. At inception, the portfolio has zero net exposure to SOL price movements, so the LP expects to earn from fees rather than directional bets. However, as in options, this neutrality is not preserved dynamically. As the price deviates from the entry level, exposure reappears due to gamma, which in derivatives terminology refers to the rate at which delta changes with respect to the underlying price (see [5]). For this reason, the strategy is often described as short gamma. The LP collects fees while holding a position whose value curve is concave in price. Inpractice, this means earning yield in calm markets but suffering losses when volatility increases, i.e., harvesting fees in exchange for bearing convexity risk (see [4]).

2. Model setup

We consider the pair SOL/USDC, where the pool price P denotes the number of USDC required to buy one unit of SOL. The LP allocates capital into a finite band [p_ℓ, p_u] within which liquidity is active. This feature allows capital to be concentrated where trading is expected, increasing fee yield relative to the same notional deployed in a constant-product AMM (see [1, 2]).

If LP contributes liquidity units L over the band [p_ℓ, p_u], then at any price P [p_ℓ, p_u] the position consists of a specific number of SOL tokens x and USDC tokens y. These amounts are determined by the geometry of the Uniswap v3 liquidity curve:

$$x=L\times(\frac{1}{\sqrt{P}} - \frac{1}{\sqrt{p_{u}}} ) \quad\quad\quad\quad\quad\text{(2.1)}$$

$$y=L\times(\frac{1}{\sqrt{P}} - \frac{1}{\sqrt{p_{l}}} ) \quad\quad\quad\quad\quad\text{(2.2)}$$

Equation (2.1) states that the SOL holdings shrink as price rises toward p_u, while equation (2.2) shows that USDC holdings grow as price rises above p_ℓ. Intuitively, inside the band the portfolio is a continuously rebalanced mixture of SOL and USDC, and outside the band it collapses into a single asset (all SOL below p_ℓ, all USDC above p_u). The dollar value of the position at price P is then:

$$V=P\times x+y \quad\quad\quad\quad\quad\quad\quad\text{(2.3)}$$

This is the gross notional, i.e., the market value of the position before accounting for leverage or debt. Let D_SOL denote the borrowed SOL (marked at price P ) and D_USDC the borrowed USDC. The equity value of the LP is, therefore, the net of asset value minus debts:

$$E=V-(D_{SOL}\times P+D_{USDC}) \quad\quad\quad\text{(2.4)}$$

Equation (2.4) makes explicit that leverage increases V relative to the LP’s own capital, but simultaneously introduces repayment obligations that increase with price in the case of SOL borrowing.

To measure LP’s exposure, we compute the sensitivity of equity to small changes in the price of SOL. Differentiating equation (2.4) with respect to P yields:

$$\Delta=\frac{\delta E}{\delta {P}}=\frac{\delta V}{\delta {P}}-D_{SOL}\quad\quad\quad\quad\text{(2.5)}$$

Expanding the derivative of V as:

$$\frac{\delta V}{\delta {P}}=\frac{\delta(P\times x+y)}{\delta {P}}=x+P\times\frac{\delta x}{\delta {P}}+\frac{\delta y}{\delta {P}}=x-P\times\frac{L}{2\times P^{3/2}}+\frac{L}{2\times P^{1/2}}=x \quad\text{(2.6)}$$

Therefore, the equity delta is:

$$\Delta=\frac{\delta E}{\delta {P}}=x-D_{SOL}\quad\quad\quad\quad\text{(2.7)}$$

This equation shows that the LP’s net dollar exposure comes entirely from the difference between SOL held through the CLMM position and SOL owed due to leverage. The LP’s position is delta-neutral if and only if these two quantities offset, i.e., D_SOL = x. Inother words, the LP borrows exactly as much SOL as the CLMM position holds at the current price P . At that moment, small upward or downward moves in SOL have no first-order effect on equity. This mirrors the construction of delta-neutral hedge strategy in option theory, but here the mechanism is implemented directly on-chain by combining CLMM with borrowed assets.

3. Numerical example

To make the abstract formulas concrete, let us work through a detailed numerical example of how a pseudo-delta-neutral position is constructed inside a CLMM. We take the SOL/USDC pair as a case study. All values are in USDC unless otherwise noted.

• Current spot price: P₀ = 200 USDC per SOL.

• Price band: [p_ℓ, p_u] = [190, 210] (a ±5% band around spot).

• Equity capital (own funds): E₀ = 100.

• Target leverage: λ = 3, so total gross notional exposure is N₀ = 300 and borrow amount is 200.

Step 1. Liquidity calibration. The first step is to find how many units of CLMM liquidity L must be minted so that the gross value of the pool position equals the target N₀ = 300 at the entry price P₀ = 200. The position value at price P is:

$$V=P\times x+y=L \times (2 \times \sqrt{P}-\sqrt{p_{l}}-\frac{ P}{\sqrt {p_{u}}})$$

Substituting P = 200, p_ℓ = 190, p_u = 210:

$$V=L\times(2\times\sqrt{200}-\sqrt{190}-\frac{ 200}{\sqrt {210}})=300$$

gives L ≈ 429.6. Once L is known, we can calculate the actual asset composition at entry using equations (2.1) and (2.2):

$$x≈0.73\quad SOL, \quad\quad y≈153.9\quad USDC$$

This means that the CLMM deposit holds about 0.73 SOL and 154 USDC at the entry price. The combined market value is 300, as desired.

Step 2. Debt allocation for neutrality. The leverage comes from borrowing assets. Here, the LP borrows 200 of notional, while contributing 100 of their own funds. To achieve delta-neutrality at inception, the borrow amount is structured so that the SOL debt offsets the SOL held inside the CLMM. Since x = 0.73 SOL, we set D_SOL = x = 0.73. This ensures that the LP is short in SOL (via debt) exactly the same amount that the CLMM position is long in SOL. The remaining debt capacity is taken in USDC to reach the total borrow notional of 200:

$$D_{USDC}= 200 − P \times D_{SOL} = 200 − 200 \times 0.73 ≈ 54$$

Thus, the liabilities are short in 0.73 SOL and short in 54 USDC.

Step 3. Equity check at entry. At P₀ = 200, the asset side is 0.73 SOL (146 USDC) plus 153.9 USDC, i.e., 300. The liability side is 0.73 SOL debt (146 USDC) plus 54 USDC debt, i.e., 200. Equity is E = 300 − 200 = 100, which matches the LP’s own capital. Moreover, the delta is ∆ = x − D_SOL = 0.73 − 0.73 = 0, so the position is exactly delta-neutral at inception.

4. Case analysis - price evolution

Now let us examine what happens as the price moves. The CLMM composition changes automatically, while the debt obligations evolve linearly with price (for the SOL debt) or stay constant (for the USDC debt). We compute both equity and net exposure in three cases.

Case A. Price unchanged (P = 200). Nothing changes. Equity remains 100, and delta is still 0. This is the ideal point, the LP has no directional exposure and collects fees while neutral.

Case B. Price falls to P = 190 (the lower bound). At the lower band edge, all capital is converted into SOL as:

$$x≈ 1.52, \quad\quad y=0,\quad\quad V≈ 288.5$$

Debt obligations are 0.73 · 190 + 54 = 193.7. Thus, the equity is:

$$E = 288.5 − 193.7 = 94.8$$

Equity has dropped below the starting 100, a sign of impermanent loss. The net SOL exposure is:

$$x − D_{SOL} = 1.52 − 0.73 = 0.79 \quad SOL$$

meaning the LP is now long nearly 0.8 SOL unhedged. The intuition behind this is as price falls, the CLMM makes you accumulate more SOL, but the SOL debt remains fixed, so you end up net long.

Case C. Price rises to P = 210 (the upper bound). At the upper edge, all capital is converted into USDC:

$$x=0, \quad\quad y≈303.7,\quad\quad V≈ 303.7$$

Debt is 0.730 · 210 + 54 = 207.3. Hence, the equity is:

$$E = 303.7 − 207.3 = 96.4$$

Equity again drops below 100. Net SOL exposure is:

$$x − D_{SOL} = 0 − 0.73 = −0.73\quad SOL$$

This means that the LP is now net short SOL, since they owe SOL but no longer hold any inside the CLMM.

Case D. Liquidation. Once price leaves the band, the CLMM position freezes. Thus, below 190 it holds a constant x_const in SOL, above 210 it holds a constant y_const in USDC. The downside liquidation occurs below the lower edge when the amount held in SOL. Thus,the LP always holds 1.525 SOL once price falls past 190, but continues to owe 0.73 SOL. The danger of liquidation is when the equitydrops below the required maintenance threshold. With a maintenance margin θ = 0.8, the critical downside price is:

$$P_{liq,↓}=\frac{D_{USDC}}{\theta\times x_{const}-D_{SOL}} ≈ \frac{54}{0.8\times 1.525-0.73} ≈ 110$$

Therefore, liquidation occurs near P = 110. This is far below the entry, but not impossible if a sharp bear move occurs.

Upside liquidation occurs above the upper edge when the position is frozen in USDC: In this region, the LP holds only USDC ≈ 303.7, while still owing D_SOL = 0.73 SOL. As the price of SOL rises, the USDC value of this SOL liability increases linearly as D_SOL x P. Liquidation occurs when the total debt reaches the maintenance fraction θ of assets, the critical upside price is:

$$P_{liq,↑}=\frac{\theta \times y_{const}-D_{USDC}}{D_{SOL}} ≈ \frac{0.8\times303.7-54}{0.73} ≈ 259$$

Thus, on the upside, liquidation would occur if the SOL price rises above roughly 259. In practice, equity starts eroding as soon as P > 210 because the position is frozen in USDC while the SOL liability grows. Here, proactive rebalancing is advisable well before the liquidation boundary.

At first sight, one might conclude that upside liquidation risk is negligible. However, this interpretation is misleading, because liquidation is only the terminal boundary where equity reaches the maintenance margin. In practice, the position becomesunattractive long before that point due to equity erosion and leverage amplification.

• Equity erosion: When P > p_u = 210, the CLMM position is frozen in USDC (y_const ≈ 303.7). The debt obligations, however, still include a fixed short position of D_SOL = 0.73 SOL. As the SOL price increases, the USDC value of this liability growslinearly, while the asset side remains constant. Consequently, equity declines steadily with every dollar increase in P .

• Leverage spikes: Leverage is defined as gross notional over equity. Once equity falls, leverage rises even if the debt is unchanged. In this example, at P = 250 the equity has dropped to 66.7, while the gross position is still around 303. Effective leverage, therefore, jumps from 3 at inception to more than 4.5 at P = 250. As P increases further, leverage escalates rapidly, amplifying both funding costs and sensitivity to adverse moves.

Note that liquidation prices should not be read as safe buffers. They only indicate the extreme boundary where a protocol enforces closure. From a risk management perspective, LPs should monitor equity erosion and leverage ratios continuously and take action (close, repay debt, or re-center liquidity) as soon as equity drawdown becomes material. In leveraged CLMM strategies, waiting until liquidation is triggered is rarely rational, because by then most of the LP’s initial capital has already been consumed by debt growth. Rebalancing well before the theoretical liquidation point is, therefore, essential to preserve LP’s capital.

Figure 4.1: Pseudo-delta-neutral position

Figure 4.1 depicts equity trajectory of the leveraged CLMM position across different price scenarios. It illustrates three cases: inside the active range (190 P 210), below the lower bound (P < 190) where assets are frozen in SOL, and above the upper bound (P > 210) where assets are frozen in USDC. As the price deviates from the entry point P₀ = 200, equity gradually erodes. On the downside, liquidation risk emerges around P 110, while on the upside it emerges around P 259. The slope of the curve beyond the active range corresponds to the constant exposure in the frozen asset versus the debt liability. This visualization makes explicit how equity evolves and why rebalancing is required long before formal liquidation.

5. Interpretation

The leveraged CLMM construction is conceptually equivalent to a delta-hedged derivatives position in classical finance. In option theory, a portfolio is delta-neutral when the first-order sensitivity of its value to the underlying vanishes. This is achieved by taking an offsetting position in the underlying asset against options exposure. Similarly, in the CLMM, the neutrality at inception is achieved by borrowing precisely the amount of SOL that equals the pool’s SOL holdings, so that ∆ = 0 at the entry price P₀. However, delta neutrality in this context is only local. The curvature of the equity function E with respect to price is governed by the gamma of the CLMM position. Differentiating the equity delta in equation (2.7) with respect to the price P yields the gamma:

$$\Gamma =\frac{\delta\Delta}{\delta {P}}=\frac{\delta x}{\delta {P}}=-\frac{L}{2}\times P^{-3/2}<0 \quad for \quad P∈(pℓ,pu) \quad\text{(5.1)}$$

In derivatives terminology, gamma measures how quickly delta changes as the price moves (see [5]). A portfolio with positive gamma adjusts its sensitivity to price movements in a way that favors the investor. If the price rises, the portfolio’s delta increases, meaning the position becomes more long and captures additional upside. If the price falls, the delta decreases and becomes more negative, so the position shifts short and gains from the downside move. In both directions, the portfolio automatically tilts in the direction of the prevailing trend. In contrary, a portfolio with negative gamma adjusts in the opposite, disadvantageous way. When prices rise, its delta turns increasingly short, so the position loses more as the upward price movement persists. When prices fall, its delta turns increasingly long, so the position accumulates losses as the decline deepens. Negative gamma, therefore, erodes neutrality and makes the portfolio vulnerable to volatility in either direction.

In CLMM, a negative gamma means that the equity curve E as a function of price is concave as depicted in figure 4.1. It peaks atthe entry price P₀ and bends downwards on both sides. This concavity explains why the position is only neutral at inception. As soon as the price drifts away from P₀, the delta shifts, positive below P₀, negative above P₀—and the position accumulates directional exposure. The directional consequences are:

• For P < P₀, the CLMM amount x increases as the pool converts into more SOL, while the SOL debt remains constant. The equity delta, therefore, becomes positive, leaving the LP net long SOL and exposed to further downside moves.

• For P > P₀, the CLMM amount x decreases and eventually vanishes, but the SOL debt remains fixed. The equity delta becomes negative, leaving the LP net short SOL and exposed to further upside rallies.

• Outside the range [p_ℓ, p_u], the CLMM amounts freezes into a constant token vector (x_const, 0) or (0, y_const). The net exposure then plateaus at a constant long or short SOL position, and the liquidation threshold depends on the interaction between this frozen position and the debt vector (D_SOL, D_USDC). This dependence has been formalized in continuous-time models as a solvency condition linking leverage, band width, and volatility (see [4]).

In summary, a pseudo-delta-neutral leveraged CLMM strategy is not riskless. It mirrors a delta- hedged but short-gamma derivatives position, meaning, locally hedged at inception, maximized at the entry price, but increasingly vulnerable to volatility as the underlying drifts. Fee income must be sufficient to compensate for both funding costs and the expected convexity drag, or the strategy becomes unprofitable.

References

[1] Adams, H., Zinsmeister, N., Salem, M., Keefer, R., & Robinson, D. (2021). Uniswap v3 Core. White paper.

[2] Elsts, A. (2021). Liquidity Math in Uniswap v3. Technical Note.

[3] Elsts, A., & Klas, K. (2023). Concentrated Liquidity with Leverage. Working paper.

[4] Tung, S.-N., & Wang, T.-H. (2024). A Mathematical Framework for Modelling CLMM Dynamics in Continuous Time. Working paper.

[5] Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.