💰 quantitative finance

Equilibrium Liquidity and Risk Offsetting in Decentralised Markets

This paper establishes a structural equilibrium framework for decentralized exchanges, demonstrating that liquidity providers strategically adjust market depth to manage risk through a trade-off involving risk aversion, replication costs, and private information, which collectively determine the economic viability and profitability of liquidity provision.

Original authors: Fayçal Drissi, Xuchen Wu, Sebastian Jaimungal

Published 2026-03-05

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Fayçal Drissi, Xuchen Wu, Sebastian Jaimungal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a bustling digital marketplace called a Decentralized Exchange (DEX). It's like a giant, automated vending machine where people can swap digital assets (like cryptocurrencies) without a middleman. But for this machine to work, someone needs to stock the shelves. That "someone" is the Liquidity Provider (LP).

This paper asks a simple but crucial question: Is it actually profitable and safe for someone to stock these shelves, especially when they are competing with a giant, centralized supermarket (the CEX) next door?

The authors build a complex economic model to answer this, but here is the story in plain English, using some creative analogies.

The Main Characters

The Liquidity Provider (The Shopkeeper): This is the person putting money into the DEX vending machine. They want to earn fees from people trading, but they are terrified of losing money if the price of the goods changes while they are holding them.
The Arbitrageurs (The Price Police): These are fast traders who instantly fix any price differences between the DEX vending machine and the big Centralized Exchange (CEX) supermarket. If the DEX is too cheap, they buy there and sell at the supermarket. This keeps prices aligned but costs the Shopkeeper money (this is called "adverse selection").
The Noise Traders (The Shoppers): Regular people who just want to buy or sell because they need to, not because they know the future price. They pay fees to the Shopkeeper.
The Centralized Exchange (The Supermarket): The big, liquid market where the Shopkeeper can go to "hedge" (protect) themselves if they get stuck holding too much inventory.

The Three-Act Play

The paper models this as a game played in three stages:

Act 1: The Shopkeeper Sets the Shelves (Liquidity Depth)
Before anyone trades, the Shopkeeper decides how much stock to put on the shelves.

The Dilemma: If they put more stock, they attract more shoppers and earn more fees. But, if the price of the goods swings wildly, they lose more money to the "Price Police" (Arbitrageurs).
The Twist: The Shopkeeper knows they can run to the Supermarket (CEX) to sell their extra stock if the price drops. But running to the supermarket costs money (trading fees) and takes effort.

Act 2: The Hedge (Risk Management)
Once the shelves are set, the Shopkeeper watches the market. If the price of the goods starts to drop, they rush to the Supermarket to sell their DEX inventory to avoid a loss.

The Cost of Running: The Supermarket isn't free. Every time the Shopkeeper runs back and forth to balance their books, they pay a "tax" (trading costs).
The Risk Aversion: If the Shopkeeper is very scared of losing money (high risk aversion), they will run to the Supermarket very aggressively to keep their exposure at zero. If they are bold, they might stay put and take the risk.

Act 3: The Shoppers Arrive
Finally, the shoppers arrive. They look at how full the shelves are.

Deep Shelves: If the shelves are full (high liquidity), the price doesn't change much when they buy. They are happy to trade big amounts.
Empty Shelves: If the shelves are thin, the price jumps around. Shoppers trade less.

The Big Discoveries (The "Aha!" Moments)

The paper finds some surprising things about how this game plays out:

1. The "Shelf Size" is a Safety Tool
Usually, people think the only way to manage risk is to run to the Supermarket (hedge). But this paper says: The Shopkeeper can also manage risk by simply putting less stock on the shelves.

Analogy: If you are scared of getting wet in the rain, you can either carry a heavy umbrella (hedge in the Supermarket) or just stay inside (reduce liquidity). The paper shows that Shopkeepers often choose to stay inside (reduce liquidity) if the cost of the umbrella is too high.

2. The "Fear vs. Cost" Ratio
The decision of how much to hedge depends on a specific ratio: How scared are you? vs. How expensive is the umbrella?

If you are terrified of risk but the Supermarket is cheap to visit, you will hedge aggressively.
If you are terrified but the Supermarket is expensive, you will actually reduce your liquidity (stock less) to avoid having to run there so much.
Key Insight: Paradoxically, being more risk-averse can lead to less liquidity in the market, because the Shopkeeper is too scared to hold the inventory that requires expensive hedging.

3. The "Crystal Ball" Effect (Private Information)
What if the Shopkeeper has a crystal ball (private information) that tells them the price will go up or down tomorrow?

Small Predictions: If the prediction is small, the Shopkeeper gets excited. They think, "I can make a quick profit on the side!" so they add more stock.
Big Predictions: If the prediction is huge (e.g., "The price is going to crash!"), the Shopkeeper gets scared. They realize that if they hold stock, the "Price Police" will drain them, and the cost of running to the Supermarket to fix it will be massive. So, they remove stock to avoid the disaster.
Result: Having inside information doesn't always make the market deeper; sometimes it makes it thinner because the Shopkeeper gets too scared to participate.

The Bottom Line

This paper explains why Decentralized Exchanges sometimes feel "thin" (hard to trade big amounts) or why liquidity disappears during volatile times.

It's not just about fees. It's a delicate balance between:

The Fees (the reward).
The Risk (losing money to price swings).
The Cost of Protection (how expensive it is to hedge in the big market).

If protecting yourself is too expensive, or if you are too scared of the risk, the Shopkeeper will simply close the shop (reduce liquidity) rather than risk a loss. The model shows that for these markets to thrive, the cost of hedging needs to be low, and the Shopkeepers need to feel that the fees are worth the risk of holding the inventory.

1. Problem Statement

The paper addresses the economic viability of liquidity provision in Decentralised Exchanges (DEXs) operating alongside Centralised Exchanges (CEXs). While existing literature often treats liquidity supply and trading conditions as exogenous and assumes frictionless risk replication, this paper argues that these factors are endogenous.

The core problem is to determine the optimal liquidity depth ( $\kappa$ ) a Liquidity Provider (LP) should supply in a DEX when:

Adverse Selection: Arbitrageurs align DEX prices with fundamental values, causing predictable losses (Loss-Versus-Rebalancing or LVR) for the LP.
Costly Risk Management: LPs must hedge their inventory risk by trading in a CEX, but this replication incurs execution costs (price impact) and is subject to the LP's risk aversion.
Private Information: LPs may possess private signals regarding future price drifts, which influence both speculative opportunities and anticipated hedging costs.
Strategic Interaction: Liquidity takers (noise traders) adjust their trading volumes based on the DEX's liquidity depth and associated trading costs.

The authors seek to characterize the equilibrium where the LP balances fee revenue against adverse selection costs, inventory risk, and the costs of dynamic hedging.

2. Methodology

The authors formulate the problem as a three-stage sequential game solved via backward induction within a partial equilibrium framework.

Stage 3: Noise Liquidity Takers (LTs)

Agents: Price-sensitive noise traders arrive stochastically (Poisson process).
Mechanism: LTs maximize utility based on private valuation shocks, balancing the benefit of trading against execution costs (slippage due to convexity of the DEX pricing curve and proportional fees).
Result: Optimal trading volume is derived as a function of the liquidity depth $\kappa$ . Higher $\kappa$ reduces slippage, increasing trade size and fee revenue for the LP.

Stage 2: LP's Dynamic Hedging in CEX

Agents: The LP manages inventory risk by trading the risky asset in a CEX.
Objective: Maximize terminal wealth minus a penalty for deviations from perfect replication (zero net inventory exposure) and minus execution costs.
Model:
- State: DEX reserves ( $Y_t$ ) and CEX inventory ( $Q_t$ ).
- Dynamics: Fundamental price $F_t$ follows a geometric Brownian motion with a stochastic drift driven by a private signal $A_t$ .
- Costs: Linear price impact in the CEX ( $\eta \nu_t$ ) and a quadratic penalty ( $\frac{\phi}{2}(Q_t + Y_t)^2$ ) for inventory risk.
Mathematical Tool: The problem is solved using variational calculus and stochastic control. The LP's objective is shown to be a linear-quadratic functional. The optimal strategy is derived as the solution to a coupled system of Forward-Backward Stochastic Differential Equations (FBSDEs).
Key Insight: The optimal strategy decomposes into:
1. Replication Component: Driven by the ratio of risk aversion ( $\phi$ ) to trading costs ( $\eta$ ).
2. Speculative Component: Driven by the private signal $A_t$ .

Stage 1: LP's Liquidity Supply Decision

Decision: The LP chooses the initial liquidity depth $\kappa$ (and thus initial reserves) to maximize expected Profit and Loss (PnL).
Trade-off: The LP anticipates the Stage 3 fee revenue (increasing in $\kappa$ ) and the Stage 2 costs (LVR and hedging costs, increasing in $\kappa$ ).
Existence: The authors prove the existence of an equilibrium $\kappa^*$ for general convex trading functions.

Extension: Transient Price Impact

The model is extended to a setting where LP trading in the CEX causes transient price impact (prices deviate temporarily and decay). This reduces the LP's hedging problem to a Differential Riccati Equation (DRE), ensuring a unique and computable solution.

3. Key Contributions

Endogenization of Liquidity: Unlike prior works, this paper endogenizes the size of liquidity supply. It demonstrates that liquidity depth is not just a passive parameter but a primary risk management instrument.
Risk-Offsetting Ratio: The authors identify the ratio of risk aversion to replication costs ( $\psi = \sqrt{\phi/2\eta}$ $ψ = ϕ /2 η$ ) as the critical determinant of hedging aggressiveness.
- High $\psi$ leads to aggressive replication, reducing risk but increasing trading costs.
- To mitigate these costs, the LP optimally reduces liquidity supply ex ante.
Non-Monotonic Effect of Private Information:
- Moderate Drifts: Private information generates speculative gains, encouraging higher liquidity.
- Large Drifts: Anticipated large price movements imply massive future reserve adjustments and high hedging costs. This "defensive" reaction leads to thinner markets (lower liquidity).
Closed-Form Solutions: For Constant Product Markets (CPMs, e.g., Uniswap), the authors derive explicit closed-form expressions for equilibrium liquidity, decomposing it into a "fundamental" component (fees vs. adverse selection) and a "speculative" component.

4. Key Results & Findings

Liquidity as a Risk Tool: The LP manages risk primarily by adjusting the scale of liquidity rather than just the intensity of hedging. When hedging costs are high or risk aversion is strong, the LP shrinks the pool size to limit exposure.
Determinants of Viability:
- Risk Aversion vs. Costs: A higher ratio of risk aversion to trading costs reduces equilibrium liquidity and mean profitability but lowers return variance.
- Volatility: Higher fundamental volatility increases adverse selection costs, reducing equilibrium liquidity.
- Trading Costs: Higher CEX trading costs ( $\eta$ ) reduce the scale of liquidity provision but do not alter the per-unit risk-return profile (returns scale down with liquidity).
Private Information Asymmetry: In Constant Product Markets, the mapping from price to reserves is convex. Therefore, negative price signals (price drops) cause larger reserve expansions than positive signals cause contractions. Consequently, negative private signals induce a stronger defensive reduction in liquidity than positive signals induce an increase.
Equilibrium Conditions: Liquidity provision is viable only if the profitability parameter (fees and demand intensity) exceeds a threshold determined by fundamental volatility ( $\gamma > \sigma^2/4$ ). If this condition fails, the DEX shuts down.

5. Significance

Theoretical Advancement: The paper bridges the gap between market microstructure theory and DeFi mechanics by incorporating costly hedging and strategic liquidity provision. It moves beyond the "frictionless" assumption of earlier DEX models.
Practical Implications for DEX Design:
- It suggests that DEX protocols relying on passive LPs may suffer from liquidity contraction during periods of high volatility or when LPs have strong private information about adverse price moves.
- It highlights the importance of the relationship between DEX liquidity depth and CEX market depth; if CEX hedging is too expensive, DEX liquidity will naturally dry up.
Risk Management: The findings challenge the notion that LPs can simply "hedge away" risk. Instead, the cost of hedging fundamentally alters the supply curve of liquidity, making liquidity depth a strategic variable for risk control.
Policy/Protocol Design: The results imply that mechanisms reducing CEX hedging costs (e.g., better integration between DEX and CEX) or mitigating adverse selection (e.g., dynamic fees, MEV protection) are crucial for sustaining deep and liquid DEX markets.

In summary, the paper provides a rigorous structural framework showing that the viability of DEX liquidity is a delicate balance between fee revenue, adverse selection, and the cost of risk management, with liquidity depth serving as the primary lever for LPs to manage this trade-off.