The Wealth of Liquidity Providers in Cl Pools

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

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The wealth of liquidity providers in CL pool

In this section, we consider a strategic LP who dynamically tracks the marginal rate Z. In our model, the LP’s position is self-financed throughout an investment window [0, T], so the LP repeatedly withdraws her liquidity and collects the accumulated fees, then uses her wealth, i.e., the collected fees and the assets she withdraws, to deposit liquidity in a new range. In the remainder of this work, we work in a filtered probability space Ω, F, P; F = (Ft)t∈[0,T] that satisfies the usual conditions, where F is the natural filtration generated by the collection of observable stochastic processes that we define below.

\ We assume that the marginal exchange rate in the pool (Zt) t∈[0,T] is driven by a stochastic drift (µt) t∈[0,T] and we write

\ dZt = µt Zt dt + σ Zt dWt , (5)

where the volatility parameter σ is a nonnegative constant and (Wt)t∈[0,T] is a standard Brownian motion independent of µ. We assume that µ is cadl ` ag with finite fourth moment, i.e., ` E [µ 4 t ] < +∞ for 0 ≤ t ≤ T. The LP observes and uses µ to optimise her liquidity positions and improve trading performance.

\ Consider an LP with initial wealth x˜0, in units of X, and an investment horizon [0, T], with T > 0. At time t = 0 , she deposits quantities (x0, y0) in the range Z ℓ 0 , Zu 0 , so the initial depth of her position is κ˜0, and the value of her initial position, marked-to-market in units of X, is x˜0 = x0 + y0 Z0. The dynamics of the LP’s wealth consist of the value of the liquidity position in the pool (αt) t∈[0,T] , the fee revenue (pt) t∈[0,T] , and the rebalancing costs (ct) t∈[0,T] .

\ We introduce the wealth process (˜xt = αt + pt + ct)t∈[0,T] , which we mark-to-market in units of the reference asset X, with x˜0 > 0 known. At any time t, the LP uses her wealth x˜t to provide liquidity. Next, Subsection 3.1 studies the dynamics of the LP’s position α in the pool, Subsection 3.2 studies the dynamics of the LP’s fee revenue p, and Subsection 3.3 studies the dynamics of the rebalancing costs c.

\ 3.1. Position value

In this section, we focus our analysis on the position value α. Throughout the investment window [0, T], the holdings (xt , yt) t∈[0,T] of the LP change because the marginal rate Z changes and because she continuously adjusts her liquidity range around Z. More precisely, to make markets optimally, the LP controls the values of δ ℓ t t∈[0,T] and (δ u t ) t∈[0,T] which determine the dynamic liquidity provision boundaries Z ℓ t t∈[0,T] and (Z u t ) t∈[0,T] as follows:

   (Z u t ) 1/2 = Z 1/2 t / (1 − δ u t /2), Z ℓ t 1/2

= Z 1/2 t 1 − δ ℓ t /2 , (6)

where δ ℓ ∈ (−∞, 2], δ u ∈ [−∞, 2), and δ ℓ δ u/2 < δℓ +δ u because 0 ≤ Z ℓ < Zu < ∞. We define δ ℓ and δ u in (6) in terms of √ Z to simplify and linearise the CL constant product formulae; see Cartea et al. (2023b) for more details. In practice, the LP earns fees when the rate Zt is in the LP’s liquidity range (Z ℓ t , Zu t), so

δ ℓ ∈ (0, 2], δ u ∈ [0, 2), and δ ℓ δ u/2 < δℓ + δ u . 6 Below, Section 4 considers a problem where the controls are not constrained, and values δ ℓ ∈/ (0, 2], δ u ∈/ [0, 2) are those where liquidity provision is unprofitable.

\ In the remainder of this paper we define the spread δt of the LP’s position as

δt = δ u t + δ ℓ t , (7)

and we consider admissible strategies δ that are R-valued and such that R T 0 δ −4 t dt < ∞, almost surely; see Section 4 for more details. For small position spreads, we use the first-order Taylor expansion to write the approximation

Z u t − Z ℓ t . Zt = (1 − δ u t /2)−2 − (1 − δ ℓ t /2)2 ≈ δt .

We assume that the marginal rate process (Zt) t∈[0,T] follows the dynamics (5). Cartea et al. (2023b) show that the holdings in assets X and Y in the pool for an LP who follows an arbitrary strategy Z ℓ t , Zu t are given by

xt = δ ℓ t δ ℓ t + δ u t αt and yt = δ u t Zt δ ℓ t + δ u t αt , (8)

so the value (αt) t∈[0,T] of her position follows the dynamics

\ s dαt = ˜xt 1 δ ℓ t + δ u t − σ 2 2 dt + µt δ u t dt + σ δu t dWt = dPLt + ˜xt 1 δ ℓ t + δ u t (µt δ u t dt + σ δu t dWt) , (9)

\ where the predictable and negative component PLt = − σ 2 2 R t 0 x˜s δs ds is the PL of the LP’s position which scales with the volatility of the marginal rate. PL is related to LVR (see Milionis et al. (2022)) which decomposes the loss of LPs in traditional CFMs into a hedgeable market risk and an unhedgeable component due to profits made by arbitrageurs. The dynamics in (9) show that a larger position spread δ reduces PL and the overall risk of the LP’s position in the pool; see Cartea et al. (2023b) for more details.

\ For a fixed value of the spread δt = δ ℓ t + δ u t , the dynamics in (9) show that if µt ≥ 0, then the LP increases her expected wealth by increasing the value δ u , i.e., by skewing her range of liquidity to the right. However, note that the quadratic variation of the LP’s position value is d⟨α, α⟩t = ˜x 2 t σ 2 (δ u t /δt) 2 dt , so skewing the range to the right also increases the variance of the LP’s position.

\ On the other hand, if µ ≤ 0, then the LP reduces her losses by decreasing the value of δ u or equivalently increasing the value of δ ℓ , i.e., by skewing her range of liquidity to the left. Thus, the LP uses the expected changes in the marginal rate to skew the range of liquidity and to increase her expected terminal wealth.

\ 3.2. Fee income

==3.2.1. Fee income: pool fee rate==

The dynamics of the fee income in our model of Section 4 uses a fixed depth κ and assumes that the pool generates fee income for all LPs at an instantaneous pool fee rate π; clearly, these fees are paid by LTs who interact with the pool, see (1)–(2). The value of π represents the instantaneous profitability of the pool, akin to the size of market orders and their arrival rate in LOBs.

\ In contrast to the proportional fee τ , which represents the fixed percentage applied to the trade sizes of LTs to compute the fees paid to the pool, the pool fee rate π denotes the instantaneous profitability of the pool computed as the percentage of total fees paid by LTs relative to the size of the pool.

\ Thus, the pool fee rate π depends on the intensity of the liquidity taking flow, the size of the pool, and the proportional fee τ . Below, we also consider the fee revenue p of one LP who maximises her wealth by choosing an optimal spread for her liquidity position.

\ To analyse the dynamics of the pool fee rate π, we use historical LT transactions in Uniswap v3 as a measure of activity and to estimate the total fee income generated by the pool; Appendix A describes the data and Table A.4 provides descriptive statistics. Figure 1 shows the estimated fee rate π in the ETH/USDC pool. For any time t, we use7

πt = 0.05% Vt 2 κ Z1/2 t ,

where Vt is the volume of LT transactions the day before t, 2 κ Z1/2 t is the pool value in terms of asset X at time t, and 0.05% is the fixed fee of the pool.8 Figure 1 suggests that the pool fee rate π generated by liquidity taking activity in the pool is stochastic and mean reverting. Here, we assume that π is independent of the rate Z over the time scales we consider; Table 1 shows that the pool fee rate is weakly correlated to the rate Z at different sampling frequencies, especially for the higher frequencies we consider in our numerical tests. In Section 4, the pool fee rate π follows Cox-Ingersoll-Ross-type dynamics.

\ ==3.2.2. Fee income: spread and concentration risk==

In the three cases of (3), increasing the spread reduces the depth κ˜ of the LP’s position. Recall that the LP fee income is proportional to κ/κ, ˜ where κ is the pool depth. Thus, decreasing the value of κ˜ potentially reduces LP fee income. Figure 2 shows the value of κ˜ as a function of the spread δ. However, although narrow ranges increase the potential fee income, they also increase concentration risk; a wide spread (i.e., a lower value of the depth κ˜) decreases fee income per LT

transaction but reaps earnings from a larger number of LT transactions because the position is active for longer periods of time (i.e., it takes longer, on average, for Z to exit the LP’s range). Thus, the LP must strike a balance between maximising the depth κ˜ around the rate and minimising the concentration risk, which depends on the volatility of the rate Z. In our model, the LP continuously adjusts her position around the current rate Z, so we write the continuous-time dynamics of (4), conditional on the rate not exiting the LP’s range, as

\ dpt = (˜κt / κ) | {z } Position depth πt |{z} Fee rate 2 κ Z1/2 t | {z } Pool size dt , (10)

where (˜κt)t∈[0,T] models the depth of the LP’s position and p is the LP’s fee income for providing liquidity with depth κ˜ in the pool. The fee income is proportional to the pool size, i.e., proportional to 2 κ Z1/2 t . Next, use the second equation in (3) and equations (6)–(8) to write the dynamics of the LP’s position depth κ˜t as

κ˜t = 2 ˜xt 1 δ ℓ t + δ u t Z −1/2 t ,

so the dynamics in (10) become

dpt = 4 δ ℓ t + δ u t πt x˜t dt . (11)

In practice, there is latency in the market and the LP cannot reposition her liquidity position and rebalance her assets continuously. Thus, the LP faces concentration risk in between the times she repositions her liquidity; narrow spreads generate less fee income because the rate Z may exit the range of the LP’s liquidity, especially in volatile markets.

\ To model the losses due to concentration risk in the continuous-time dynamics (11) of the LP’s fee revenue, we introduce an instantaneous concentration cost that reduces the fees collected by the LP as a function of the spread; the concentration cost increases (decreases) when the spread narrows (widens). We modify the dynamics of the fees collected by the LP in (11) as follows

dpt = 4 δ ℓ t + δ u t πt x˜t dt − γ 1 δ ℓ t + δ u t 2 x˜t dt , (12)

where γ > 0 is the concentration cost parameter and x˜t is the wealth invested by the LP in the pool at time t. To justify the form of the concentration cost, we study the realised fee revenue in the ETH/USDC pool as a function of the spread δ in (7) for rebalancing frequencies m = 1 minute and m = 5 minutes. We denote by pbδ,m the average realised fee revenue earned by an LP in the ETH/USDC pool who provides liquidity with wealth x˜ = 1, in a range with spread δ, around the marginal rate Z throughout a time window of length m.

\ The left panel of Figure 3 shows that the value of δ which maximises the fee revenue pbδ,m is strictly positive for both values of m that we consider. In particular, narrow ranges reduce fee revenue due to concentration risk. The form of the fee revenue dynamics (12) is a second order Taylor approximation that captures the specific shape of the fee revenue in CL markets; see left panel of Figure 3. The LP uses the regression model δ 2 pbδ,m = 4 π m δ − γ m ,

\ which is based on the dynamics (12), to estimate the concentration cost parameter γ. The right panel of Figure 3 shows that δ 2 pbδ,m is affine in δ and that the estimated concentration cost parameter γ depends on the rebalancing frequency of the LP. In particular, the figure shows that the dynamics (12) and the second order approximation are suitable to describe the realised fee revenue in CL markets. The performance study of Section 5 uses this methodology to set the value of the concentration cost parameter γ.

\ ==3.2.3. Fee income: drift and asymmetry==

The stochastic drift µ indicates the future expected changes of the marginal exchange rate in the pool. In practice, the LP may use a predictive signal so that µ represents the belief that the LP holds over the future marginal rate. For an LP who maximises fee revenue, it is natural to consider asymmetric liquidity positions that capture the liquidity taking flow. We define the asymmetry of

a position as

ρt = δ u t / δ u t + δ ℓ t = δ u t /δt , (13)

where δ u t and δ ℓ t are defined in (6). In one extreme, when the asymmetry ρ → 0, then Z u → Z and the position consists of only asset X, and in the other extreme, when ρ → 1, then Z ℓ → Z and the position consists of only asset Y. In the remainder of this work, the asymmetry of the LP’s position is a function of the observed drift:

ρt = ρ (δt , µt) = 1 2 + µt δt = 1 2 + µt δ u t + δ ℓ t , ∀t ∈ \[0, T\] . (14)

\ The asymmetry (14) adapts the skew of the position to the expected drift of the marginal rate. When µ = 0, the position is symmetric around the marginal rate and ρt = 1/2, so δ ℓ t = δ u t . When µ > 0, the position is skewed to the right (i.e., δ u t > δℓ t ) to capture more LT trades and fee revenue, and similarly when µ < 0, the position is skewed to the left (i.e., δ u t < δℓ t ).

\ Also, when µ > 0 and the position is skewed to the right according to (14), the proportion of asset Y increases and the position profits from rate appreciation, and when the position is skewed to the left, the proportion of asset Y decreases and the position is protected from rate depreciation.

\ Optimal liquidity provision is the dynamic choice of δ u and δ ℓ over a trading window, or equivalently, the dynamic choice of δ and ρ. Our model assumes that the LP fixes the asymmetry ρ of her position at time t according to (14), so we reduce the trading problem to a one-dimensional dynamic optimisation problem, which significantly simplifies calculations.

\ To further justify the form of the asymmetry (14), we use Uniswap v3 data to study how the asymmetry and the width of the LP’s range of liquidity relate to fee revenue. First, we estimate the realised drift µ in the pool ETH/USDC over a rolling window of T = 5 minutes.9 Next, for any time t, the fee income for different positions of the LP’s liquidity range is computed for various values of the spread δ and for various values of the asymmetry ρ.

\ For each value of the realised drift µ during the investment horizon, and for each fixed value of the spread δ, we record the asymmetry that maximises fee income. Figure 4 shows the optimal (on average) asymmetry ρ as a function of the spread δ of the position for multiple values of the realised drift µ.

\ Figure 4 suggests that there exists a preferred asymmetry of the position for a given value of the spread δ and a given value of the drift µ. First, for all values of the spread δ, the LP skews her position to the right when the drift is positive (ρ ⋆ > 0.5) and she skews her position to the left when the drift is negative (ρ ⋆ < 0.5).

\ Second, for narrow spreads, the liquidity position requires more asymmetry than for large spreads when the drift is not zero. In our liquidity provision model of Section 4, the LP holds a belief over the future exchange rate throughout the investment window and controls the spread δ = δ u + δ ℓ of her position. Thus, she strategically chooses the asymmetry of her position as a function of δ and µ. We approximate the relationship exhibited in Figure 4 with the asymmetry function (14).

\ 3.3. Rebalancing costs and gas fees

Our model considers a strategic LP who continuously repositions her liquidity position in the pool which requires rebalancing of the LP’s assets. Specifically, repositioning typically leads to different holdings (8) in the pool, in which case the LP trades one asset for the other in the pool or in another trading venue. However, rebalancing assets to reposition liquidity is costly.

\ Let (ct) t∈[0,T] denote the cost of rebalancing in terms of asset X. Similar to Fan et al. (2021) and Fan et al. (2022), we model rebalancing costs as proportional to the quantity yt of asset Y that the LP deposits in the pool. At any time t, we assume that the LP uses all her wealth x˜t when repositioning her liquidity position, so we use (8) to write

ct = −ζ yt Zt = −ζ δ u t δ ℓ t + δ u t x˜t , c0 = 0 , (15)

where ζ > 0 is a constant that models the execution costs. Transactions sent to the pool also bear gas fees. Gas fees are a flat fee paid to the blockchain and do not depend on the size of the LP’s transaction; see Lı et al. (2023) for more details on the impact of gas fees on liquidity provision. Thus, gas fees scale with the frequency at which the LP sends transactions to the pool.

\ Our model considers continuous trading, so gas fees do not influence the optimisation problem, but should be considered when the strategy is implemented. In the next section, we derive an optimal liquidity provision strategy, and prove that the profitability of liquidity provision is subject to a tradeoff between fee revenue, PL, and concentration risk.

:::info Authors:

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

:::

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

:::

\