This post attempts to organize a basic understanding of liquidity, oriented at DeFi. We’ll define liquidity and its key metrics, and analyze it for constant-product AMMs. Then we’ll touch on impermanent loss and end with a brief overview of concentrated liquidity.

Liquidity

Liquidity is a property of markets, crucial for capital efficiency. Folklore intuition can be summed up in one line:

A market is liquid if it can quickly execute trades with minimal price changes.

For example, major stock exchanges are liquid markets for commonly traded stocks.

It is typical to abuse terminology and say an asset is liquid, meaning “this asset is quickly convertible into money with minimal loss”. Formally, this statement asserts the liquidity of an implicit market between the asset and money. For examples, a house is an illiquid asset because it takes a while both to discover the market price and to sell it at that price.

One sometimes encounters the term “funding liquidity”, referring to the ease of obtaining credit (i.e., borrowing). Thus an asset has high funding liquidity if it can serve as collateral for money at a good rate.

For some history see e.g. this stackexchange answer.

Liquidity is measurable by a few key metrics:

Metric	Meaning	Consequence	Cause
Spread	Lowest sell price (best ask) minus highest buy price (best bid).	Wide spread → high cost to initiate trade	Insufficient market-making
Depth	What asset amount can be traded within some price range?	Shallow depth → volatility	Insufficient supply
Price impact	How does my trade impact the current market state?	High price impact → I get less	App-level: depth & spread
Slippage	What I expected when placing my trade vs what I got when it executed.	High slippage → I get less	Core-level: market state discrepancy between simulation & execution, e.g. due to competition, latency, MEV

Spread is a threshold cost which dominates small trades. Larger trades begin to consume depth, which outweighs the initial spread.

Before diving a bit deeper, we record a common unit of measurement in finance:

Definition. A basis point (plural bps) is a hundredth of a percent $0.01％=\frac 1{10,000}$.

Spread

Assuming the market price is within spread interval, a taker must pay a premium to immediately initiate a trade. Wider spread → higher premium.

\[[\text{highest bid},\text{lowest ask}]\]

A narrow spread thus increases capital efficiency, especially for small trades. A spread of 10 bps means the highest bid and lowest ask differ by $0.10％$.

Depth

High depth around an exchange rate leads to price stability: even large trades will not cause volatility. The statement “there’s 10K USD of BTC buy-side depth per bp at the current price” means you can buy 10K USD-worth of BTC and incur a price change of at most one basis point, i.e $0.01％$.

The depth of liquidity in a price interval should be thought of as the supply of capital deployed there. High depth at exorbitantly high/low prices is wasted capital. We’ll unpack this later for constant-product AMMs.

Depth is crucial also for stable capital-efficient loan markets because it ensures liquidations are profitable to liquidators. We illustrate the opposite extreme: how shallow depth triggers a liquidation death spiral. The loan market in question is borrowing USD against BTC collateral.

• Suppose Alice loaned 1M USD to Bob against 10 BTC. Suppose Bob defaults, so that Alice seizes 10 BTC. Now suppose Alice wishes to liquidate the loan i.e. sell the BTC for USD. If the BTC/USD market in question has shallow liquidity, the exchange rate would dip against Alice: she may sell 1 BTC at a good price, but any larger amount would sell at an increasing loss to her. In case of an over-collateralized loan, Alice can make a quick profit even if she sells some of her BTC underpriced. Such a sale would lower the BTC/USD exchange rate against BTC, potentially triggering more liquidations.
• Depth is the dampener of liquidations, defending against chain-reactions and death spirals.

Price impact

Products sold in stores typically have posted prices, which trivialize economic calculation. The naive approach to buying some amount of an asset would be to multiply its market price-per-unit by the amount.

If the market price is a 1 USD/unit, I naively expect 100 USD to buy me 100 units. Price impact is any deviation from this expected outcome. Such deviations are deterministic in the current market state, and determined entirely by trade size, spread, and depth. Price impact can be mitigated at the application level, e.g. via aggregators and routers which traverse multiple markets.

Slippage

Even if I have perfect knowledge of the market mechanism (e.g. AMM curve) and its current state, I do not know the future market state at the execution of my order. Despite my ability to perfectly simulate my trade, I cannot predict the market conditions at execution: reality can differ from my expectations, resulting in slippage. Such slippage can be honest, e.g. competitors pay higher priority fees and overtake me, or manipulative, e.g. sequencers manipulating ordering.

Market-makers

Markets are not in a constant state of equilibrium, and organic activity (or lack thereof) can result in occasional wide spreads.

1. Buyers and sellers need not show up together. There can be waves of one-sided demand, in terms of both time and volume.
2. Price discovery takes time: buyers would initially post low bids and sellers would initially post high asks.
3. Liquidity may not arise organically, due to wide spreads or shallow depth.

Market makers (henceforth MMs) streamline market operations by providing supply & demand counterparties and optimizing price discovery.

Order-book market-makers

In order-book markets, MMs act by posting trades that bring the market closer to equilibrium. The naive business model is simple: manufacture spread and profit off it. Specifically, MMs manufacture spread by maintaining a gap between their highest bids and lowest asks. They aim for a narrow spread when competition is high and risk is low, and conversely. Let’s make this a bit more explicit. Assume Alice and Bob are competing MMs. If Alice quotes (posts orders) slightly tighter than Bob, then her orders are more attractive to traders, and they’ll get executed first. This is obviously good if she profits, but a volatile market may actually cause tight spread to incur losses.

Although MMs profit off of wider spread, a competitive market also incentivizes them to provide depth around their tightest quotes. Indeed, once demand depletes Alice’s tightest quotes, she no longer has any competitive advantage over Bob. The more depth she provides, the more orders she will be able to capture from Bob at her tighter spread (despite profiting less per unit exchanged).

To summarize, order-book market-makers are precisely liquidity providers!

AMMs; impermanent loss

An AMM is an implementation of a market that:

1. Automatically computes exchange rates based on its state.
2. Automatically distributes supply across the exchange-rate curve to create liquidity.

Since AMMs automatically manage capital, the only required making is the deployment of capital. It so happens that AMM terminology refers to these capitalists as liquidity providers. As we shall see below, more refined AMM designs actually involve makers who choose where to deploy capital, i.e. where to create depth. In this sense, they are indeed liquidity providers as opposed to “mere capitalists”.

Before continuing, we’ll define an important concept that is often overcomplicated by only looking at examples.

Definition. A pair (LP position, initial deposit) is at impermanent loss if the current the value of the LP position is lower than the current value of the initial deposit (i.e. at current exchange rates).

To emphasize - impermanent loss is only defined with respect to an initial deposit; it’s not an intrinsic property of an LP position. Put succinctly, impermanent loss is the opportunity cost of being an LP instead of holding on to the capital in a wallet. This opportunity cost if offset by expected profit from trading fees.

Impermanent loss is so-called because it may be transient. It is caused by arbitrage traders reacting to market shifts:

1. The market outside the AMM shifts, affecting the market exchange rate between the AMM assets.
2. Arbitrage traders buy the appreciated asset from the AMM for cheaper than on the external market.
3. The AMM supply of the appreciated asset decreases until the AMM discovers market price.
4. If an LP redeems their share of each asset, they end up with a smaller amount of the appreciated asset than they deposited, but a larger amount of the depreciated asset.
5. The current value of the redeemed LP position is lower than the current value of the original deposit.

Risk of impermanent loss vanishes once AMM exchange rates return to the neighborhood of the initial deposit. For this reason, it is customary for volatile pair LPs to charge higher fees.

The above formulation should make it clear there is no “impermanent gain” - arbitrage traders buy whichever asset is undervalued by the AMM.

Worked example: constant-product AMMs

We now illustrate the abstract nonsense above in the setting of a constant-product AMM. We start out without fees, and address them later.

Basics

Fix two assets $X,Y$ and denote by $x,y$ their respective supplies. The AMM curve is $xy=k$ for some constant $k$. Evidently the curve is symmetric in $x,y$. As a function of $x$, we see $y=\tfrac kx$ is not only monotone decreasing but also subadditive, which enforces diminishing marginal utility. Conceptually, this is the only required property of an AMM curve. In our constant-product case, marginal diminution is visualized by the hyperbola curve: As the supply $x$ grows, a fixed change of supply $\Delta x$ causes a diminishing change of supply $\Delta y$.

Let us quantify supply changes. Suppose the present state of AMM supply is $x,y$. How does a change in supply $\Delta x$ affect the supply of $Y$? By definition, the next state of the AMM supply is $x+\Delta x, y+\Delta y$, so the invariant constraint gives us

\[k=(x+\Delta x) (y+\Delta y)=y(x+\Delta x)+\Delta y(x+\Delta x).\]

Hence

\[\Delta y=\frac{k-y(x+\Delta x)}{x+\Delta x}=\frac{k}{x+\Delta x}-y=\frac{-y\Delta x}{x+\Delta x}.\]

Thus we see how $\Delta y$ inversely depends on the initial supply $x$.

The exchange rate (price) charged by the AMM is just the ratio of supplies. The fraction $\frac{y}{x}$ is the supply of $Y$ fitting into a unit supply of $X$. That is, the $Y$-price of a unit of $X$. Price change is easily computable:

\[\frac{y}{x}=\frac{k}{x^2}\longrightarrow\frac{y+\Delta y}{x+\Delta x}=\frac{k}{(x+\Delta x)^2}.\]

Note the $Y$-price of a unit of $X$ obeys an inverse square law w.r.t to the supply of $X$: price decays quadratically as supply grows. This super-linear decay also exhibits diminishing marginal utility.

Prices are not encoded by the AMM equation, but rather discovered by the market. The equation merely constrains the supplies to satisfy diminishing marginal utility, which is necessary for price discovery. (Play around with a constant-sum equation and see how the resulting market is severely misbehaved. Hint: the exchange ratio must be 1:1, so no diminishing marginal utility and therefore no price discovery and the pool is easy to drain.)

Depth analysis

In this section we’ll quantitatively analyze the depth in constant-product AMMs. Our goal is to answer questions like “what percent of the supply is employed for trading within a given price range”.

We can express supply $x$ as a function of the $Y$-price per unit of $X$: $p=\frac yx$.

\[x^2p=k\implies x(p)=\sqrt \frac{k}{p}.\]

Note a larger constant $k$ means more depth: the market sustains a larger supply of $X$ at a given price.

By definition, depth is the supply change required to move between given prices.

\[D_X(p_1\to p_2)=x(p_1)-x(p_2)=\sqrt\frac k{p_1}-\sqrt\frac k{p_2}\]

Depth is nonnegative iff $p_1\leq p_2$: supply decreases iff price increases (in this AMM). Evidently depth is antisymmetric, so the price interval determines a unique number - the absolute value of the associated depth. Note also how depth scales with the interval:

\[\text{Depth}[\alpha^2 p_1,\alpha^2 p_2]=\frac 1\alpha \text{Depth}[p_1, p_2].\]

Constant-product AMM depth decays slowly as $\frac1\surd$.

Equipped with the depth formula, we can analyze the capital efficiency of constant-product AMMs. Specifically, we’ll analyze how the constant product curve causes supply to be inefficiently allocated to price bands with low trading activity.

Depth on the tail

Assume a current supply of $x_0$ at a price-per-unit of $p_0$. This supply is distributed over the price range $[p_0,\infty)$ precisely according to depth.

• A divergent increasing sequence of prices $p_0<p_1<p_2<\cdots$ partitions the interval $[p_0,\infty)$.
• Current supply is partitioned by the depths of the associated price bands. (The infinite sum makes sense because $x(p)\overset{p\to \infty}{\longrightarrow}0$.)

\[\begin{aligned} x_0 &=[x(p_0)-x(p_1)]+[x(p_1)-x(p_2)]+\cdots \\ &=D_X(p_0\to p_1)+D_X(p_1\to p_2)+\cdots \end{aligned}\]

The partition of total current supply by depth specifies the amount of current supply allocated to each price band.

How much supply is “wasted” on exorbitantly high prices? Consider a $\overbrace{\text{USDC}}^Y/\overbrace{\text{ETH}}^X$ pool with $k=10^{10}$. As of July 2025, 1 ETH ≈ 3000 USD, so $p_0=3000$ whence $x_0=\sqrt\frac{10^{10}}{3000}\approx 1825$. A high USD-price of ETH is denoted $p_\top$. The depth formula gives

\[D_X(p_\top\to \infty)=x(p_\top)-x(\infty)=\sqrt\frac k{p_\top}.\]

Some example computations show the extreme capital inefficiency of the constant-product curve.

\[\begin{aligned} D_{\text{ETH}}(\mathrm{\$}1\mathrm{M} \to \infty) &\approx 100 \text{ ETH} &\approx 5.5％ \\ D_{\text{ETH}}(\mathrm{\$}10\mathrm{K} \to \infty) &\approx 1000 \text{ ETH} &\approx 55％ \\ D_{\text{ETH}}(\mathrm{\$}5\mathrm{K} \to \infty) &\approx 1414 \text{ ETH} &\approx 77％ \end{aligned}\]

Depth in general

How to interpret $D_X(p_1\to p_2)$ if the current price $p_\text{now}$ satisfies $p_\text{now}>p_1$? In this case, reaching $p_1$ would require an increase in supply, so we can’t use the partition trick above. Or can we? Math shows the way. The equation

\[D_X(p_1\to p_2)=D_X(p_1\to p_\text{now})+D_X(p_\text{now}\to p_2)\]

interprets the LHS as the amount of supply required for a price increase $p_1\nearrow p_\text{now}$, plus the (familiar) depth starting from the current price.

Depth concentration

We can write an explicit formula for the relative depth concentration in a price band within a larger band $[p_1,p_2]\subset [p_\perp,p_\top]$, notably independent of the constant $k$.

\[\underset{[p_1,p_2]\subset[p_\perp,p_\top]}{\text{Concentration}}=\frac{\text{Depth}[p_1,p_2]}{\text{Depth}[p_\perp,p_\top]}=\frac{\sqrt\frac 1{p_1}-\sqrt\frac 1{p_2}}{\sqrt\frac 1{p_\perp}-\sqrt\frac 1{p_\top}}\]

Let’s analyze the capital efficiency of a constant-product AMM for a stable pair USDC/USDT. by looking at the relative concentration of a bp interval vs a 1% interval (factor of 100)

\[\begin{aligned} \underset{[0.9999,1.0001]\subset[0.9,1.1]}{\text{Concentration}} &\approx 0.0994％ \\ \underset{[0.99,1.01]\subset[0.9,1.1]}{\text{Concentration}} &\approx 9.94％ \end{aligned}\]

So assuming the entire market lives within a 10% band, merely 10% of capital is allocated to trading within a %1 band, and merely 1% is allocated to trading within a basis point!

Impermanent loss

Consider an ETH/USDC pool with initial supply of 100 ETH and 100K USDC, so initial ETH price is 1000 USDC and the constant is $k=10^7$. Now assume ETH price doubled to 2000 USD on the outside. Traders flock in to buy the underpriced ETH from the AMM. We compute the ETH reserves at the end of price discovery, when the AMM price has stabilized at 2000 USDC per ETH:

\[x(p)=\sqrt\frac kp=\sqrt \frac{10^7}{2000}=\sqrt{5000}\approx 70.71.\]

Suppose Alice initially owned 10% of the pool.

• Her initial deposit was 10 ETH and 10K USDC. Current value = 30,000 USD.
• Her current LP position is 7.071 ETH and 14,142 USDC. Value ≈ 28,284 USD.

What about fees?

Percentage fees are crucial for LP revenue. They slightly increase price impact, especially for larger trades.

Concentrated liquidity

Classically, there is a clear distinction between capitalists, who passively provide capital, and those who actively employ capital (often delegated to them). In DeFi terminology, capitalists are the passive liquidity providers. However, active liquidity providers are superior market makers, who have strong potential to enhance capital efficiency. We illustrate this with a case study: Uniswap v2 vs Uniswap v3.

Uniswap v2 is a constant-product AMM. LP UX is roughly: choose pair → deposit assets → receive LP token → earn fees → exit. LPs must supply equal value of both tokens (to preserve pool price). Since LP positioned are fully determined by deposited capital, LP tokens are just fungible ERC-20 tokens.

Uniswap v3 is a “piecewise-constant-product” AMM. LP UX is roughly: choose pair → choose fee tier → choose price range → deposit assets → receive LP token → earn fees → reposition/exit. Each pool partitions the allowed price range via ticks, via which LPs can specify their desired price bands for deploying capital (adding depth). Due to the added complexity of the liquidity position, LP tokens are (currently) NFTs, encoding more than the ERC-20 standard allows.

The ability to concentrate capital in a chosen price range precisely means LPs are free to choose where to provide depth. Let’s illustrate how beneficial this design is for capital efficiency by examining a stable pair, e.g. USDC/USDT (I wonder how well this will age). The vast majority of LPs will choose to provide depth in a narrow band about the 1:1 rate, so liquidity will be deep in the center and shallow at de-pegged exchange rates. If Alice deposits in tick range $[t_1,t_3]$ and Bob deposits in $[t_2,t_4]$, both are active in $[t_2,t_3]$, and both provide liquidity to the constant-product AMM associated to this small interval. The locality of each piece in Uniswap v3 mitigates the slow $\frac1\surd$ decay of depth in constant-product AMMs to small intervals.

Unfortunately the piecewise design has a significant drawback: it exacerbates the risk of impermanent loss. For example, assume Alice concentrates her capital at in the range $[2900,3100]$ USD per ETH. Suppose ETH begins tp appreciate with respect to USD in the outside market. As usual, arbitrage traders buy the strong asset (ETH) from the pool. However, since Alice concentrated all her capital in this narrow range, if ETH appreciates beyond the interval, her entire initial deposit of ETH would be depleted and exchange for USD due to arbitrage. In other words, the narrow price range makes Alice more susceptible to impermanent loss. As usual, if the price moves back in range, the impermanent loss vanishes, and Alice can continue her plans for world domination.

In a sense, piecewise-constant-product AMMs interpolate between a constant-product AMM on one end, and an order-book in the other: liquidity providers act as market makers by choosing where to deploy capital. Concentrated liquidity assists price discovery due to aligned incentives: LPs earn trading fees when their deployed capital is used, so they naturally want to provide depth near market prices. Order-books still have some advantages: market-makers have room for more capital efficient strategies, and they’re not exposed to impermanent loss!