Adaptive Fee Model

A model designed to free users from hefty fees while transacting.

Considering the CPF RA×RB=k{R_A}\times{R_B}=k, liquidity providers always incur a loss named impermanent loss, due to a state deviation of reserves. The loss will disappear if the state returns to the initial state. Without loss of generality, let’s assume that BB outperforms AA after a period. Clearly, arbitrageurs will sell AA to get BB because the current pool is maintaining an undervalued price of BB.

(24) (RA,RB)0<α1(1αRA,αRB)(R_A,R_B)\xrightarrow{0<\alpha\leq1}(\frac{1}{\alpha}R_A,{\alpha}R_B).

Thus, the price is transforming from p=RB/RAp=R_B/R_A to p=α2RB/RAp'=\alpha^2R_B/R_A. Compared with a HODL strategy, adding liquidity to a pool seems less profitable (without a fee model).

(25) lossnofee=(pRA+RB)(p1αRA+αRB)loss_{no-fee}=(p'R_A+R_B)-(p'\frac{1}{\alpha}R_A+\alpha{R_B})

(26) =(1α)2RB.=(1-\alpha)^2R_B.

Most AMMs use a fee model typically of 0.25% to cover the loss. The mistake here is the fee is fixed while the loss is varied. It’s better when the fee is adaptive, that means the fee will be large if the loss is large, and vice versa.

To create a more effective fee model, the AMM develops a function γ=f(α)\gamma=f(\alpha), where 0<γ10<\gamma\leq1, respects to α\alpha to adapt the change. Because α\alpha describes how the pool state deviates from the current state, it’s reasonable for the fee function to rely on α\alpha.

Definition 2. The impermanent loss is zero when the adaptive fee is following:

(27) γ=1α2α\gamma=\frac{1-\alpha}{2-\alpha}.

Proof. Recall that a trader will get (1α)RB(1-\alpha)R_B in a no-fee model. When the AMM applies the adaptive fee model, the trader will receive (1γ)(1α)RB(1 − \gamma)(1 − \alpha)R_B. In summary,

(28) (RA,RB)0<α,γ1(1αRA,αRB+γ(1α)RB)(R_A,R_B)\xrightarrow{0<\alpha,\gamma\leq1}(\frac{1}{\alpha}R_A,{\alpha}R_B+\gamma(1-\alpha)R_B).

With p=α(α+γ(1α))RB/RAp'=\alpha(\alpha+\gamma(1-\alpha))R_B/R_A, the loss now is

(29 )lossadaptivefee=(pRA+RB)(p1αRA+α+γ(1α))RB)loss_{adaptive-fee}=(p'R_A+R_B)-(p'\frac{1}{\alpha}R_A+\alpha+\gamma(1-\alpha))R_B)

(30) =((α2)(α+γ(1α))+1)RB=((\alpha-2)(\alpha+\gamma(1-\alpha))+1)R_B​.

Because of the zero loss, we have an equation, lossadaptivefee=0loss_{adaptive-fee}=0​. Hence, γ=(1α)/(2α).\gamma=(1-\alpha)/(2-\alpha).

Corollary 1. Applying the adaptive fee, the pool’s state will follow:

(31) (RA,RB)0<α1(1αRA,12αRB).(R_A,R_B)\xrightarrow{0<\alpha\leq1}(\frac{1}{\alpha}R_A,\frac{1}{2-\alpha}R_B).

Because the pricing curve is the same as the 2D CPF, the impermanent loss (IL) is obvious to Sentre’s AMM too. However, employing a variant of the adaptive fee model can cover the IL while also benefit LPs.

(32) Protocol Fee=Adaptive Fee+0.05%Protocol\ Fee=Adaptive\ Fee+0.05\%

The approach can secure income for LPs. By flexibility, the fee is low corresponding to small price changes, and gets larger along with the price change.

Furthermore, the AMM collects a fractional fee as a tax to maintain the core engine at the beginning. In the future, this tax can be opened as a grant for community projects (see Fig. 4).

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