How Constant Function Markets Price Trades Under Concentrated Liquidity

Table Of Links

Abstract

1. Introduction

2. Constant function markets and concentrated liquidity

• Constant function markets
• Concentrated liquidity market

3. The wealth of liquidity providers in CL pools

• Position value
• Fee income
• Fee income: pool fee rate
• Fee income: spread and concentration risk
• Fee income: drift and asymmetry
• Rebalancing costs and gas fees

4. Optimal liquidity provision in CL pools

• The problem
• The optimal strategy
• Discussion: profitability, PL, and concentration risk
• Discussion: drift and position skew

5. Performance of strategy

• Methodology
• Benchmark
• Performance results

6. Discussion: modelling assumptions

• Discussion: related work

7. Conclusions And References

Constant function markets and concentrated liquidity

2.1. Constant function markets

Consider a reference asset X and a risky asset Y which is valued in units of X. Assume there is a pool that makes liquidity for the pair of assets X and Y , and denote by Z the marginal exchange rate of asset Y in units of asset X in the pool. In a CFM that charges a fee τ proportional to trade size, the trading function f q X, qY = κ 2 links the state of the pool before and after a trade is executed, where q X and q Y are the quantities of asset X and Y that constitute the reserves in the pool, κ is the depth of the pool, and f is increasing in both arguments.

\ We write f q X, qY = κ 2 as q X = φκ q Y for an appropriate decreasing level function φκ. We denote the execution rate for a traded quantity ±y by Z˜ (±y), where y ≥ 0. When an LT buys y of asset Y , she pays x = y × Z˜ (y) of asset X, where

f (q X + (1 − τ ) x, qY − y ) = κ 2 =⇒ Z˜(y) = φκ (q Y − y ) − φκ(q Y ) (1 − τ ) y . (1)

Similarly, when an LT sells y of asset Y , she receives x = y × Z˜ (−y) of asset X, where

f (q X + x, qY + (1 − τ ) y ) = κ 2 =⇒ Z˜(−y) = φκ (q Y ) − φκ (q Y + (1 − τ ) y ) y . (2)

\ In CL markets, the marginal exchange rate is Z = −φ ′ κ (q Y ), which is the price of an infinitesimal trade when fees are zero, i.e., when y → 0 and τ = 0. 2 In traditional CPMs such as Uniswap v2, the trading function is f(q X, qY ) = q X ×q Y , so the level function is φκ(q Y ) = κ 2/qY and the marginal exchange rate is Z = q X/qY . Liquidity provision operations in CPMs do not impact the marginal rate, so when an LP deposits the quantities x and y of assets X and Y , the condition Z = q X/qY = (q X + x)/(q Y + y) must be satisfied; see Cartea et al. (2022, 23b).

\ 2.2. Concentrated liquidity market

This paper focuses on liquidity provision in CPMs with CL. In CPMs with CL, LPs specify a range of rates (Z ℓ , Zu ] in which their assets can be counterparty to liquidity taking trades. Here, Z ℓ and Z u take values in a finite set {Z 1 , . . . , ZN }, the elements of the set are called ticks, and the range (Z i , Zi+1] between two consecutive ticks is a tick range which represents the smallest possible liquidity range; see Drissi (2023) for a description of the mechanics of CL.3

\ The assets that the LP deposits in a range (Z ℓ , Zu ] provide the liquidity that supports marginal rate movements between Z ℓ and Z u . The quantities x and y that the LP provides verify the key formulae

   x = 0 and y = ˜κ Z ℓ −1/2 − (Z u ) −1/2 if Z ≤ Z ℓ , x = ˜κ Z 1/2 − Z ℓ 1/2

and y = ˜κ Z −1/2 − (Z u ) −1/2 if Z ℓ < Z ≤ Z u , x = ˜κ (Z u ) 1/2 − Z ℓ 1/2 and y = 0 if Z > Z u (3)

\ where κ˜ is the depth of the LP’s liquidity in the pool. The depth κ˜ is specified by the LP and it remains constant unless the LP provides additional liquidity or withdraws her liquidity. When the rate Z changes, the equations in (3) and the prevailing marginal rate Z determine the holdings of the LP in the pool, in particular, they determine the quantities of each asset received by the LP when she withdraws her liquidity.

\ Within each tick range, the constant product formulae (1)–(2) determine the dynamics of the marginal rate, where the depth κ is the total depth of liquidity in that tick range. To obtain the total depth in a tick range, one sums the depths of the individual liquidity positions in the same tick range; see Drissi (2023). When a liquidity taking trade is large, so the marginal rate crosses the boundary of a tick range, the pool executes two separate trades with potentially different depths for the constant product formula.

\ In CPMs with CL, the proportional fee τ charged by the pool to LTs is distributed among LPs. More precisely, if an LP’s liquidity position with depth κ˜ is in a tick range where the total depth of liquidity is κ, then for every liquidity taking trade that pays an amount p of fees,4 the LP with liquidity κ˜ earns the amount

p˜ = κ˜ κ p 1Zℓ (4)

Thus, the larger is the position depth κ˜, the higher is the proportion of fees that the LP earns.5 The equations in (3) imply that for equal wealth, narrow liquidity ranges increase the value of κ. ˜ However, as the liquidity range of the LP decreases, the concentration risk increases.

:::info Authors:

1. Alvaro Cartea ´
2. Fayc¸al Drissia
3. Marcello Monga

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:::info This paper is available on arxiv under CC0 1.0 Universal license.

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