Strategic Analysis of Just-In-Time Liquidity Provision in Concentrated Liquidity Market Makers

Automated Market Makers (AMMs) enable decentralized token exchanges through algorithmic pricing instead of traditional order books. We begin by explaining one of the most common classes of AMMs known as constant product market makers (CPMMs) [6]. A CPMM maintains a liquidity pool of tokenized assets $X$ and $Y$. After initialization, the CPMM maintains a pool price $q$, which is the price of token $Y$ in terms of token $X$. When an LP enters the CPMM, they contribute an amount of liquidity $L$, which comprises token amounts $x$ of $X$ and $y$ of $Y$, according to the following rule:

Consequently, the token amounts $x,y$ of each liquidity position of amount $L$ always follow the constant product rule $xy=L^2$. Additionally, when there exist multiple liquidity positions ${\{(x_i,y_i,L_i)\}}_{i=1}^N$, the corresponding summations also follow the constant product rule

A CPMM supports three basic operations – mint (add liquidity), burn (remove liquidity), and swap (trade).

• $\mathrm{■}$

  Mint and burn transactions are executed by an LP who specifies liquidity $L$ to add/remove, and will deposit/withdraw token amounts $(x,y)$ following (1). $q$ remains constant.

• $\mathrm{■}$

  Swaps are executed by a trader who specifies an amount of either token to pay. Without loss of generality, assume the trader pays $\mathrm{\Delta }x$ tokens of type $X$. $L$ remains unchanged for each liquidity position, so the trader will get $\mathrm{\Delta }y$ $Y$ tokens following (2):

  $$(x_{\text{total}}+\mathrm{\Delta }x)(y_{\text{total}}-\mathrm{\Delta }y)=L_{\text{total}}^2.$$

  As a result, the pool has a new price $q^{\prime }=(y_{\text{total}}-\mathrm{\Delta }y)/(x_{\text{total}}+\mathrm{\Delta }x)$ and the equivalent token amount $(x^{\prime },y^{\prime })$ of each liquidity position $L$ will update by plugging $q^{\prime }$ into (1).

Slippage is an important concept in this domain; it refers to the difference between the execution price of a trade and the price expected at the time of trade initiation. In the context of AMMs, slippage arises from the fact that large trades move the price along the bonding curve, resulting in a marginally worse exchange rate for each successive unit traded. Slippage is an inherent byproduct of price impact in AMMs.

Since the emergence of CPMMs, more complex AMM designs have been proposed; among the most widespread of these is Concentrated Liquidity Market Makers (CLMMs) [14]. They allow LPs to choose not only the amount of liquidity they wish to add to the pool, but also the price range in which the liquidity is active. When an LP adds liquidity to the CLMM, the LP creates a liquidity position by specifying a tuple $(L,a,b)$, where $L$ is the amount of liquidity, and $(a,b)$ represents the price range in which the position is effective. During a trade, if the AMM price exits the specified range $(a,b)$ of an LP’s position, that liquidity will not be used to support the trade outside the specified price range. At price $q$, this liquidity position reserves token amounts

where ${\widehat{q}}_{a,b}=\min \{b,\max \{a,q\}\}$ is the projection of the price $q$ onto the interval $(a,b)$. By setting $(a,b)=(0,\mathrm{\infty })$, this relation reduces to (1), which implies that a CPMM is a special case of a CLMM when every LP chooses the full price range. However, in general cases where , the reserves of the CLMM liquidity position remain constant with $x=0$ when $q\ge b$ and with $y=0$ when $q\le a$. This implies that 1) the position is unused when $q$ moves out of range $(a,b)$, and 2) there exists a maximum reserve of token $X$ and token $Y$ for such positions.

A CLMM also supports mint, burn and swap operations, as long as they are consistent with the total reserves in the pool (e.g. do not try to swap tokens that do not exist). A swap outputs $\mathrm{\Delta }y$ in the following steps:

1. 1.

   Calculate the new price $q^{\mathbf{\prime }}$: Since, for each position, $x$ is a continuous and monotonic function of $q$, the total liquidity reserve $x_{\text{total}}$ is also continuous and monotonic in $q$. Therefore, we can determine the post-swap price $q^{\prime }$ by solving for the value of $q^{\prime }$ that satisfies $x_{\text{total}}(q^{\prime })=x_{\text{total}}(q)+\mathrm{\Delta }x(q^{\prime })$, which is well-defined due to the invertibility of $x_{\text{total}}(\cdot )$. Provided that $q^{\prime }$ remains within the union of all active price ranges, the swap can be successfully executed (refer to [1] for implementation details).

2. 2.

   Update the token reserve: Each position keeps $(L,a,b)$ constant, so we recalculate $x^{\prime }$ and $y^{\prime }$ by plugging $q^{\prime }$ into (3). The change in the second token, $\mathrm{\Delta }y$, can then be obtained by summing all $y^{\prime }$ values.

By concentrating liquidity within a narrower price range, CLMMs enable each active position to support a higher volume of trades with the same amount of capital. This increased capital efficiency was designed to reduce price slippage for traders - as larger trades can be absorbed with smaller price movements [1]- and to allow LPs to extract more from their money, since they can have a higher liquidity, therefore a higher share of the fees, using the same budget by picking a narrower range. At the same time, when the market price exits a position’s specified range, that liquidity becomes inactive or “frozen”, and only the remaining active positions support the trade. As a result, traders of such trades may experience higher slippage compared to a scenario where liquidity is distributed across the entire price spectrum.

In practice, CLMMs discretize the price axis using a system of ticks. Let $T$ be the set of possible ticks, $T=\{t_i|i=0,\mathrm{\dots },M\}$, with $M\in {\mathbb{Z}}_{>0}$.³³3Ticks are defined by exponential spacing: $T=\{t_i|i=0,\mathrm{\dots },M\}$, where $t_i={1.0001}^{\tau (i-\iota )}$ with $\tau ,M\in {\mathbb{Z}}_{>0}$ and $\iota \in \mathbb{Z}$. As a result, the set of valid price ranges is given by . Liquidity providers are restricted to choosing ranges $(a,b)\in ℛ$, so liquidity remains constant between consecutive ticks $(t_m,t_{m+1})$. Within each such interval (or across consecutive intervals with identical liquidity), price changes follow the structure defined in (3), which reveals a tight coupling between price movement and the liquidity available in the pool.

Prior literature commonly assumes that the price ratio between two tokens is stable and externally set by the market, given by $q_{\text{market}}=\frac{p_x}{p_y}$, where $p_x$ and $p_y$ denote the market prices of tokens $X$ and $Y$, respectively [25, 3, 4]. However, the actual price within a CLMM pool is determined endogenously – it is only changed by trades in the pool. If it does not coincide with the market price, arbitrage activities become profitable. At equilibrium – when no arbitrage opportunities remain – the two values align, such that $q=q_{\text{market}}$. This alignment is often assumed in theoretical analyses, particularly in the absence of active participants like JIT LPs and arbitrageurs.

The divergence between the pool price $q$ and the market price thus represents both a risk and an opportunity. On one hand, price mismatches expose LPs to impermanent loss – the difference in value between a liquidity position and the equivalent token holdings outside the pool. On the other hand, such mismatches create arbitrage opportunities for sophisticated agents, including JIT LPs. To encompass both favorable and adverse outcomes of price divergence, we adopt a more general term and refer to the impermanent loss effect as price impact throughout the remainder of this paper.

Consider an LP $n$ who mints a position when the pool price is $q$ and later withdraws it at pool price $q^{\prime }$. Let $p_x$ and $p_y$ denote the external market prices of tokens $X$ and $Y$ at the time of minting, and let $p_x^{\prime }$ and $p_y^{\prime }$ be their respective market prices at the time of withdrawal. Following (3), we denote by $x_n(q)$ and $y_n(q)$ the token amounts associated with the position of LP $n$ when the pool price is $q$ (at minting), and by $x_n(q^{\prime })$ and $y_n(q^{\prime })$ the corresponding amounts when the price is $q^{\prime }$ (at withdrawal). The dollar value of the position at the time of minting and at the time of withdrawal is:

In what follows, to simplify notation we omit the explicit dependencies on parameters of the above quantities, e.g., referring to ${𝒞}_n(\mathrm{\Delta }x,L;p_x,p_y,p_x^{\prime },p_y^{\prime },q,q^{\prime })$ simply as ${𝒞}_n$. Note that a negative price impact () corresponds to a gain for the liquidity provider relative to holding the assets outside the pool. Conversely, a positive price impact implies a relative loss.

Throughout the rest of this paper, we omit the LP subscript $n$ when the context is clear, and refer to quantities such as $x_n(q)$, $y_n(q)$ simply as $x(q)$, $y(q)$.

Our first main result consists of necessary and sufficient conditions under which a price change $q\to q^{\prime }$ results in non-positive price impact. This threshold condition delineates precisely when an LP’s position becomes favorable or unfavorable relative to simply holding the tokens at market prices

We analyze the limiting behavior of price impact as available liquidity changes. Suppose we have a price range $(a,b)$ which is currently covered by total liquidity $L$ by passive LPs. A JIT LP is considering adding liquidity to this range; intuitively, when liquidity $L$ tends to zero, the position is too small to have any meaningful participation, and the price impact vanishes. Conversely, as $L\to \mathrm{\infty }$, the position becomes large enough that the price becomes almost static. Formally, we establish the following result

Unlike ordinary LPs who passively leave their liquidity in the liquidity pool, Just-In-Time (JIT) LPs react to market conditions in real-time to extract profit [26]. We next present two important aspects of JIT LP behavior, and explain how we model them.

Formally, we represent the JIT optimization problem as a tuple $([N],𝒰,𝒮,\theta ,\rho )$. Here, $[N]$ denotes the set of all passive liquidity providers in the pool. The strategy space $𝒮$ encapsulates all feasible actions available to the JIT LP, while $𝒰$ is the utility function mapping each strategy to a real-valued profit – computed as the difference between earned fees, incurred price impact, and bidding costs. The vector $\theta$ includes all relevant swap parameters, such as trade size, fee rate, price information, and the liquidity distribution of passive providers. Finally, $\rho$ is a constant that denotes the JIT budget.

Note that the JIT optimization problem assumes that the passive LP positions are established a priori. We denote by $𝒔$ the strategy profile of passive LPs, encoded as a vector of liquidity allocations across ticks: $𝒔={(L_i,a_i,b_i)}_{i\in [N]}$. The aggregate effect of the passive LP strategies $𝒔$ induces a liquidity distribution function $P={(P_m)}_{m\in M}$, where each $P_m$ denotes the total passive liquidity available at tick $m$. Formally, $P$ is derived via a deterministic mapping from $𝒔$. In the remainder of this section, we use $𝒔$ to refer to the underlying strategies and $P$ to denote the induced state of the pool.

We classify trades into three canonical scenarios – referred to as strategic archetypes – that capture the dominant patterns and analytical results of Theorems 4 and 12.

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  Overpriced trade: The trade moves the pool price away from the market price, resulting in the purchase of the more expensive token. Formally, this occurs when or $q^{\prime }>q\ge p_x/p_y$. According to Corollary 5, such movements are associated with non-positive price impact, meaning the LP benefits (Figure 2(a1)).

• $\mathrm{■}$

  Arbitrageur trade: The trade brings the pool price closer to the market price by acquiring the cheaper token. This corresponds to or $q>q^{\prime }\ge p_x/p_y$. As shown in Corollary 5, this condition implies positive price impact, i.e., a loss for LPs. Although this trade is optimal for arbitrageurs, it is typically unprofitable for JIT LPs unless the fees compensate for the loss (Figure 2(a2)).

• $\mathrm{■}$

  Overshoot trade: A corrective trade crosses the market price, starting by buying the cheaper token but pushing the pool price past the market value, ultimately overpaying. This configuration – or $q>p_x/p_y>q^{\prime }$ – lies on the boundary conditions specified in Theorem 4. In this case, whether the price impact is favorable or not depends on whether the post-trade price $q^{\prime }$ crosses the threshold condition in (9) or (10) (Figure 2(a3)).

Recall that $q$ is the pool price of token $Y$ in terms of token $X$, so $1/q$ is the pool price of $X$ in $Y$. When $q>p_x/p_y$, token $Y$ is overpriced in the pool relative to the market, implying that $X$ is relatively cheaper on-chain.

Among all archetypes, participating in overpriced trades yields the highest utility. These trades allow JIT LPs to sell tokens above market value across the entire chosen range $(a,b)$.

In overshoot trades, JIT LPs can profit by targeting only the favorable segment – i.e., by minting liquidity only in the price range where and . By concentrating liquidity narrowly in this segment, JIT LPs increase their share of fees while avoiding loss-inducing zones. Figure 2(b) gives an illustrative example; the initial and final price sandwich the market price , so the trade is “overshooting” the price. Therefore, in all ticks between $(q,p_x/p_y$) the price impact will be negative; hence the best strategy is to mint liquidity only on feasible ranges inside $(p_x/p_y,q^{\prime })$ where the price impact is positive.

In contrast, engaging in arbitrageur trades is generally unfavorable unless the fee revenue is high enough to offset the incurred price impact. The following result formally establishes a sufficient condition under which fees are insufficient to cover price impact:

The proof of the above proposition can be found in [18].

We observe three main findings from our empirical evaluation:

1. 1.

   JIT LPs in the pool we studied currently invest suboptimally and could significantly increase their profits. (§6.1.1)

2. 2.

   JIT LP utility is primarily driven by price impact rather than fees. (§6.1.2)

3. 3.

   If JIT LPs were to optimize their investment strategies, it would help JIT LPs and traders, at the expense of passive LPs. (§6.1.3)

A common perception is that arbitrageurs profit primarily at the expense of passive LPs through the price impact. However, this is not always the case. Consider a model where the market price $P_t$ evolves stochastically, and the pool price $Q_t$ follows it with some lag due to noise. Suppose a trade moves the pool price from $Q_t=P_t$ to $Q_{t+1}\ne P_{t+1}$, and an arbitrageur subsequently restores alignment at $Q_{t+2}=P_{t+2}$. If $P_t=P_{t+1}=P_{t+2}$, the passive LP’s price impact ${𝒞}_n$ is zero, but fees ${ℱ}_n>0$ are still earned from both trades.

In this case, the arbitrageur’s profit comes not from the LP, but from the noisy trader who executed the initial swap at a worse price – e.g., buying at $\sqrt{Q_t{Q}_{t+1}}>Q_t=P_t$. The arbitrageur captures this overpayment at $\sqrt{Q_{t+1}{Q}_{t+2}}$. JIT LPs benefit even more from such trades. By sandwiching the swap, they can collect both price impact gains and fees. Moreover, their presence dampens the price displacement, thereby reducing the arbitrageur’s profit and the passive LP’s fee share from the second trade.

To illustrate this, suppose a noisy trade pushes the price to $Q_{t+1}>Q_t$. A JIT LP that enters before the trade adjusts the effective spot price to $Q_{t+1}^{\ast }$, such that $Q_{t+1}>Q_{t+1}^{\ast }>Q_t$. The arbitrageur, now facing a smaller price deviation, captures less profit: .

Assuming all arbitrage opportunities are rapidly corrected, the pool price trajectory is driven by market price movements plus correctable noise. This noise does not generate price impact directly; rather, it creates temporary dislocations that can be monetized by arbitrageurs and JIT LPs. In fact, this type of exploitable noise is the basis of Loss-versus-Rebalancing (LVR) [20], a cost borne by passive LPs even when price impact vanishes.

JIT LPs exploit this noise by positioning liquidity around anticipated dislocations. Their interventions reduce price volatility, preempt arbitrage, and extract part of the fees that would otherwise accrue to passive LPs or arbitrageurs. If arbitrage profits stem from noisy trades rather than passive LPs, JIT LPs can be seen as first movers that mitigate price deviations while capturing a portion of this surplus.