Supply Control

The supply of GEX tokens is controlled by varying the liquidity product $k$ of the GLP by adding minting and burning coefficients to the AMM equations.

Mint ratio

The mint ratio of the GEX token in a GLP is calculated as the quotient between the supply ratio of the pool, $\sigma_p$, and the weight ratio of the collateral of said pool, $\omega_p$:

$$\mu_p = \frac{\sigma_p }{\omega_p}$$

The mint ratio rises when the total amount of minted tokens is below its target value or when the pool has less collateral than desired. In this case, in the event of a mint operation, the pool creates more liquidity and rises less in price compared to another GLP that has more collateral, which creates an arbitrage incentive for the pool with less collateral to attract more suppliers.

When the GLP has more collateral than desired, the mint ratio drops compared to other GLP, creating less liquidity and driving the price up more, encouraging the use of other pools to mint instead.

Burn ratio

The burn rate of GEX tokens in a pool is proportional to the pool's supply ratio $\sigma_p$ and the weight ratio $\omega_p$:

$$\rho_p = \sigma_p \cdot \omega_p$$

The difference between the burn ratio and the mint ratio is determined by the collateral weight ratio: while in burning the relationship is proportional, in minting it is the inverse. This means that the behavior of the GLP when adjusting the arbitrage incentives is opposite depending on whether it is a supply (mint) or withdrawal (burn) operation of collateral.

In a GEX token burn operation, if the pool has less collateral than desired, it burns fewer tokens which results in a lower final price, creating an arbitrage incentive to mint in the pool. If the pool instead has more collateral than desired, the amount of GEX tokens burned increases, resulting in a higher final price that discourages the supply of collateral to the pool.

Supply ratio

The supply ratio has the mission of creating liquidity for the GEX token proportionally to the amount of collateral that is deposited in the pools, thus creating a price curve with constant variance. It is calculated as a ratio between the difference of the target supply of the GEX token, $S_g$, and the total supply minted by the pools in the network $\sum_p m_p$:

$$\sigma_p = \begin{cases} \ \ \ \ \ 2 &\text{if } m_p \le 0 \\ 1 + \frac{S_g - \sum_p m_p}{S_g} &\text{if } 0 < m_p \le S_p \\ \ \ \ \ \ 1 &\text{if } m_p > S_p \end{cases}$$

The supply ratio has its range of values limited to the interval [1, 2] for stability reasons. The ratio takes its maximum value at 2 when the supply of the GEX token is null, so that the generation of liquidity is prioritized over the rise in price. When the target supply level (100 million tokens) is reached, the ratio is equal to 1, and exactly the same number of tokens that are delivered to the user are minted. When this happens, the number of GEX tokens in the pool becomes constant, so the liquidity product $k$ varies linearly with the amount of collateral in the pool, and the GEX token price curve converges to $y = x$, perfectly linear.

If the supply ratio is less than 1, the amount of GEX tokens in the pool starts to decrease and the price curve is no longer linear, approaching a parabola the closer the mint ratio gets to 0. If the supply ratio is greater than 2, an inversion of the price curve occurs: when minting GEX tokens their price would fall and when exchanging them it would rise, causing unstable behavior of the pool. For this reason the values are limited to the interval [1, 2].

Pool weight ratio

The weight ratio is responsible for creating arbitrage opportunities between the pools that help keep the amount of collateral in each one close to its target value. It is defined as the relationship between the current weight of the collateral, and the target weight set in the protocol parameters:

$$\omega_p' = \frac{v_p}{\rho_p \cdot \sum_p v_p}$$

Where $v_p$ is the value, measured in USD, of the collateral that the pool contains, $\sum_p v_p$ is the joint value of the collateral in all the GLPs in the network, and $\rho_p$ is the target weight. The value of the calculated ratio $\omega_p'$ is limited so that its value is between 0.75 and 1.25:

$$\omega_p = \begin{cases} 0.75 &\text{if } \omega_p' < 0.75 \\ \ \ \omega_p' &\text{if } 0.75 \le \omega_p' \le 1.25 \\ 1.25 &\text{if } \omega_p' > 1.25 \end{cases}$$

The weight ratio will have a value greater than 1 when the value of the collateral in the pool is greater than the target, and less than 1 when the value of the collateral is below its target.