Optimal Liquidity Provision in Concentrated AMM Pools Under Stochastic Fees

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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

Optimal Liquidity Provision In CL Pools

4.1. The problem

Consider an LP who provides liquidity in a CPM with CL throughout the investment window [0, T]. We work on the filtered probability space Ω, F, P; F = (Ft)t∈[0,T] where Ft is the natural filtration generated by the collection (Z, µ, π). The dynamics of the LP’s wealth consist of the fees earned, the position value, and the rebalancing costs.

\ Similar to Section 3.1, we denote the wealth process of the LP by (˜xt = αt + pt + ct)t∈[0,T] , with x˜0 > 0 known, where α is the value of the LP’s position, p is the fee revenue, and c is the rebalancing cost. At any time t, the LP uses her wealth x˜t to provide liquidity. Now, use (9), (12), and (15) to write the dynamics of the LP’s wealth as

\ d˜xt = ˜xt 1 δ ℓ t + δ u t 4 πt − σ 2 2 + (µt − ζ ) δ u t dt + σ δu t dWt − γ 1 δ ℓ t + δ u t 2 x˜ dt , (16)

where γ ≥ 0 is the concentration cost parameter, see Subsection 3.2, and π follows the dynamics defined below in (17). To simplify notation, we set ζ = 0; our results hold when replacing µ by µ − ζ.

Next, use δt = δ u t + δ ℓ t and δ u t / δt = ρ (µt , δt) in (16) to write the dynamics of the LP’s wealth as

d˜xt = 1 δt 4 πt − σ 2 2 x˜t dt + µt ρ (δt , µt) ˜xt dt + σ ρ (δt , µt) ˜xt dWt − γ δ 2 t x˜t dt .

Next, following the discussions of Section 3.2, we denote by η the profitability threshold and we assume that the pool fee rate π follows the dynamics

d(πt − ηt) = Γ (π + ηt − πt) dt + ψ √ πt − ηt dBt , (17)

where Γ > 0 denotes the mean reversion speed, π > 0 is the long-term mean of (πt − ηt) t∈[0,T] , ψ > 0 is a non-negative volatility parameter, (Bt) t∈[0,T] is a Brownian motion independent of (Wt) t∈[0,T] , and π0 − η0 > 0 is known. To solve the LP’s optimal liquidity provision problem, we introduce the following assumption.

\ Assumption 1 The profitability threshold η in the dynamics (17) is given by

ηt = σ 2 8 − µt 4 µt − σ 2 2 + ε 4 . (18)

From Assumption 1 and the CIR dynamics (17) it follows that

πt − ηt ≥ 0 =⇒ 4 πt − σ 2 2 + µt µt − σ 2 2 ≥ ε > 0 , ∀t ∈ \[0, T\] . (19)

\ Assumption 1 and condition (19) ensure that the spread δ of the optimal strategy is admissible. Financially, the inequality in (19) is a profitability condition πt ≥ ηt that guarantees that the LP’s fee income π is greater than the PL faced by the LP, adjusted by the drift in the marginal rate.

\ The profitability condition (19) is further discussed in Sections 4.3 and 6. Finally, note that the dynamics in (5) and (17) imply that the LP also observes W, B, and the profitability threshold η is determined by µ, so the LP observes all the stochastic processes of this problem.

\ 4.2. The optimal strategy

The LP controls the spread δ of her position to maximise the expected utility of her terminal wealth in units of X. To define the set of admissible strategies A, note that if the LP assumes that the asymmetry function ρ is that in (14), then for each δ, we need

Z T 0 ρ (δt , µt) 2 dt < ∞ P–a.s. . (20)

Observe that

Z T 0 ρ (δt , µt) 2 dt = Z T 0 1 2 + µt δt 2 dt ≤ T 4 + 1 2 Z T 0 µ 4 t dt + 1 2 Z T 0 1 δ 4 t dt .

Thus, to satisfy (20) and to ensure that the control problem below is well-posed, we define the set of admissible strategies

At = (δs)s∈\[t,T\] , R-valued, F-adapted, and Z T t 1 δ 4 s ds < +∞ P–a.s. ,

where A := A0.

Let δ ∈ A. The performance criterion of the LP is a function u δ : [0, T] × R4 → R given by

u δ (t, x, z, π, µ ˜ ) = Et,x,z,π,µ ˜ U x˜ δ T ,

where U is a concave utility function, and the value function u : [0, T] × R4 → R of the LP is

u(t, x, z, π, µ ˜ ) = sup δ∈At u δ (t, x, z, π, µ ˜ ). (21)

The following results solve the optimal liquidity provision model when the LP assumes a general stochastic drift µ and her preferences are given by a logarithmic utility function.

\ Proposition 1 Assume the asymmetry function ρ is as in (14) and that U(x) = log(x). Then,

w (t, x, z, π, µ ˜ ) = log (˜x) + (π − η) 2 Z T t Et,µ 8 2 γ + µ2 s σ 2 exp (−2 Γ (s − t)) ds (22)

+(π − η)2 Γ π + ψ 2 Z T t Et,µ \[C (s, µs)\] exp (−Γ (s − t)) ds − (π − η) Z T t Et,µ 4 ε 2 γ + σ 2 µ2 s exp (−Γ (s − t)) ds + Z T tΓ π Et,µ \[E (s, µs)\] + ψ 2 Et,µ \[ηs C (s, µs)\] ds − 1 2 Z T t Et,µ ε 2 2 γ + σ 2 µ2 s µs ds − π σ 2 8 (T − t)

solves the HJB equation associated with problem (21). Here, ηs = σ 2 8 − µs 4 µs − σ 2 2 ε 4 for s ≥ t , ηt = η, and Et,µ represents expectation conditioned on µt = µ, and

C (t, µ) = Et,µ Z T t 8 2 γ + µ2 s σ 2 exp (−2 Γ (s − t)) ds ,

and

E (t, µ) = Et,µ Z T t 2 Γ π + ψ 2 C (s, µ) + 4 ε 2 γ + σ 2 µ2 s exp (−Γ (s − t)) ds .

For a proof, see Appendix B.1.

Theorem 1 Let Assumption 1 hold and assume that the asymmetry function ρ is as in (14) and that U(x) = log(x). Then, the solution in Proposition 1 is the unique solution to the optimal control problem (21), and the optimal spread (δs) s∈[t,T] ∈ At is given by

δ ⋆ s = 2 γ + µ 2 s σ 2 4 πs − σ2 2 + µs µs − σ2 2 = 2 γ + µ 2 s σ 2 4 (πs − ηs) + ε , (23)

where ηs = σ 2 8 − µs 4 µs − σ 2 2 + ε 4 .

For a proof, see Appendix B.2.

\ 4.3. Discussion: profitability, PL, and concentration risk

In this section, we study the strategy when µ = 0 , in which case the position is symmetric ρ = 1/2 and δ u t = δ ℓ t = δt/2 and the optimal spread (23) becomes10

δ ℓ ⋆ t = δ u ⋆ t = 2 γ 8 πt − σ 2 =⇒ δ ⋆ t = 4 γ 8 πt − σ 2 , (24)

so the inequality in (19) becomes

4 πt − σ 2 2 ≥ ε > 0 , ∀t ∈ \[0, T\] . (25)

The inequality in (25) guarantees that the optimal control (24) does not explode, and ensures that fee income is large enough for LP activity to be profitable. In particular, it ensures that π > σ2/8 + ε. When ε → 0, i.e., σ 2/4 → π, the spread δ → +∞. However, we require that the spread δ = δ u + δ ℓ ≤ 4, so the conditions δ ℓ ≤ 2 and δ u ≤ 2 become

γ 4 π − σ2 2 ≤ 2 =⇒ π − γ 8 ≥ σ 2 8 . (26)

When δ ℓ = δ u = 2, the LP provides liquidity in the maximum range (0 , +∞), so the depth of her liquidity position κ˜ is minimal, the PL is minimal, and the LP’s position is equivalent to providing liquidity in CPMs without CL; see Cartea et al. (2023b) for more details. In that case, the dynamics of PL in (9) are

dPLt = − σ 2 8 αt dt ,

so σ 2/8 is the lowest rate at which the LP’s assets can depreciate due to PL. On the other hand, when δ ≤ 4, the depreciation rate of the LP’s position value in (9) is higher than σ 2/8. In particular, if δ = δ tick, where δ tick is the spread of a liquidity position concentrated within a single tick range, then the depth of the LP’s liquidity position κ˜ is maximal and PL is maximal with dynamics

dPLt = − σ 2 2 δ tick αt dt ,

so σ 2/2 δ tick is the highest rate at which the LP’s assets can depreciate due to PL. LPs should track the profitability of the pools they consider and check if the expected fee revenue covers PL before considering depositing their assets in the pool. When µ = 0, we propose that LPs use σ 2/8 as a lower bound rule-of-thumb for the pool’s rate of profitability because σ 2/8 is the lowest rate of depreciation of their wealth in the pool.

\ Condition (26) ensures that the profitability π − γ/8, which is the pool fee rate adjusted by the concentration cost, is higher than the depreciation rate of the LP’s assets in the pool. Thus, the condition imposes a minimum profitability level of the pool, so LP activity is viable. An optimal control δ ⋆ > 4 indicates non-viable LP activity because fees are not enough to compensate for the PL borne by the LP. Figure 5 shows the estimated pool fee rate and the estimated depreciation rate in the ETH/USDC pool (from January to August 2022). In particular, the CIR model captures the dynamics of πt − σ 2/8.

Next, we study the dependence of the optimal spread on the value of the concentration cost coefficient γ, the fee rate π, and the volatility σ. The concentration cost coefficient γ scales the spread linearly in (24). Recall that the cost term penalises small spreads because there is a risk that the rate will exit the LP’s range. Thus, large values of γ generate large values of the spread. Figure 6 shows the optimal spread as a function of the pool fee rate π.

\ Large potential fee income pushes the strategy towards targeting more closely the marginal rate Z to profit from fees. Lastly, Figure 7 shows that the optimal spread increases as the volatility of the rate Z increases. Finally, the optimal spread does not depend on time or on the terminal date T. The LP marksto-market her wealth in units of X, but does not penalise her holdings in asset Y .

\ In particular, the LP’s performance criterion does not include a running penalty or a final liquidation penalty (to turn assets into cash or into the reference asset). For example, if at the end of the trading window the holdings in asset Y must be exchanged for X, then the optimal strategy would skew, throughout the trading horizon, the liquidity range to convert holdings in Y into X through LT activity.11

\ 4.4. Discussion: drift and position skew

In this section, we study how the strategy depends on the stochastic drift µ. Use δt = δ ℓ t + δ u t and ρ (δt , µt) = δ u t /δt to write the two ends of the optimal spread as

δ u ⋆ t = 2 γ + µ 2 t σ 2 8 πt − σ 2 + 2 µt µt − σ2 2 + µt and δ ℓ ⋆ t = 2 γ + µ 2 t σ 2 8 πt − σ 2 + 2 µtµt − σ2 2 − µt . (27)

The inequality in (19) guarantees that the optimal control in (27) does not explode and ensures that fee income is large enough for LP activity to be profitable. The profitability condition in (26) becomes

πt − γ 8 ≥ σ 2 8 µ 2 t 2 1 − µt 4 µt − σ 2 2 ,

so LPs that assume a stochastic drift in the dynamics of the exchange rate Z should use this simplified measure of the depreciation rate due to PL as a rule-of-thumb before considering depositing their assets in the pool.

\ Next, we study the dependence of the optimal spread on the value of the drift µ. First, recall that the controls in (27) must obey the inequalities12

0 < δℓ t ≤ 2 and 0 ≤ δ u t < 2 ,

because 0 ≤ Z ℓ < Zu < ∞ and Zt ∈ (Z ℓ t , Zu t ], which together with (7) implies 0 ≤ δt ≤ 4. Next, the asymmetry function satisfies

0 < ρ (δt , µ) = δ u t δt < 1 , (28)

which implies

0 ≤ ρ (δt , µ) δt < 2 and 0 ≤ (1 − ρ (δt , µ)) δt < 2 . (29)

Now, use (14) and (29) to write

0 ≤ 1 2 + µ δt δt < 2 and 0 ≤ 1 2 − µ δt δt < 2 . (30)

Finally, use (28) and (30) to obtain the inequalities

2 |µ| ≤ δt ≤ 4 − 2 |µ| , (31)

so µ must be in the range [−1, 1] for the LP to provide liquidity. If µ is outside this range, concentration risk is too high so the LP must withdraw her holdings from the pool. Recall that the dynamics of Z are geometric and µ is a percentage drift, so values of µ outside the range [−1, 1] are unlikely. Moreover, when µ = −1, the drift of the exchange rate Z is large and negative, so the optimal range is (0, Z], i.e., the largest possible range to the left of Z. When µ = 1, the drift of the exchange rate Z is large and positive, so the optimal range is (Z, +∞), which is the largest possible range to the right of Z.

\ Condition (31) is always verified when we study the performance of the strategy in the ETH/USDC pool. Figure 8 shows how the optimal spread adjusts to the value of the drift µ. Finally, note that

∂δu ⋆ ∂σ = ∂δℓ ⋆ ∂σ = 2 µ 2 σ (4 π − 4 η + ε) + 4 σ (1 + µ) (2 γ + µ 2 σ 2 ) (4 π − 4 η + ε) 2 > 0 , ∀µ ∈ [−1, 1] ,

shows that the optimal range is strictly increasing in the volatility σ of the rate Z, which one expects as increased activity that exposes the position value to more PL, and increases the concentration risk.

:::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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