3. Token Exchange Formulas

Token exchanges within a Liquidity Bootstrapping Pool (LBP) involve two primary operations: "Swap Given In" and "Swap Given Out." These operations determine the amount of one token received when another is provided or the amount needed to receive a specified amount of another token

These operations are distinct from simple Buy/Sell transactions, as each swap involves both acquiring one token and relinquishing another. Since swaps are bi-directional, $Token_x$ and $Token_y$ can represent either $Token_{Traded}$ or $Token_{Anchor}$, depending on the direction of the swap.

3.1. Operation : ‘Swap Given In’

The ‘Swap Given In’ operation refers to the process of determining how much of a token you will receive when you input a specific amount of another token into a liquidity pool. This is typically used when a participant knows the amount they want to trade and wants to calculate how much they will receive in return.

The process begins by inputting the desired amount of tokens into the pool. The amount of tokens received is calculated based on the current token volumes and weights within the pool, using the formulas below. This calculation considers the changing dynamics within the pool, ensuring that the received amount reflects the most current state of the liquidity pool.

Because weights adjust the price sensitivity in an LBP, the ‘Swap Given In’ operation introduces a non-linear relationship to the original swap formula (without weights) from general liquidity pools; this effect is expressed by raising the volume ratio to the power of the weight ratio, $\frac{Weight_y}{Weight_x}$​​, resulting in a power-law behavior.

$Token_x^{Received}= Volume_x \times [ 1 - (\frac {Volume_y}{Volume_y + Token_y^{Input}})]^{\frac{Weight_y}{Weight_x}}$

Where

• $Token_x^{Received}$ : Amount of $Token_x$ being received from the pool (swapped out).

• $Tokeny_{Input}$ : Amount of $Token_y$ being input into the pool (swapped in).

• $Volume_x​$ : Current volume of $Token_x$ in the pool before the transaction.

• $Volume_y$​ : Current volume of $Token_y$ in the pool before the transaction.

• $Weight_x$​ : Current weight of $Token_x$ in the pool.

• $Weight_y$​ : Current weight of $Token_y$ in the pool.

The above formulas assume that the ratio between the input token volume ($Token_y^{Input}$) and the pool's existing volume of the token ($Volume_y$) remains small. As the input token volume $Token_y^{Input}$ approaches the total volume of the corresponding token in the pool ($Volume_y$), the output received ($Token_x^{Received}$) diminishes and may approach zero. This occurs because the increase in input drastically alters the pool's balance, leading to a situation where the received token amount becomes negligible. Extreme values for $Token_y^{Input}$ can cause the formula to produce outputs that are practically meaningless or extremely small, indicating that the pool is being heavily impacted by the trade.

3.2. Operation : ‘Swap Given Out’

The "’Swap Given Out’ operation involves determining how much of a token needs to be input into the pool to receive a specific amount of another token. This is used when a participant knows the exact amount of the token they want to obtain and needs to calculate the corresponding input required.

To perform a ‘Swap Given Out,’ the trader specifies the amount of the token they want to receive. The system then calculates the necessary amount of the other token that must be input into the pool based on the current volumes and weights of the tokens. This ensures the trader knows exactly how much of their existing tokens they need to commit to achieve their desired outcome.

$Token_y^{Input}= Volume_y \times [ (\frac {Volume_x}{Volume_x - Token_x^{Desired}})^{\frac{Weight_x}{Weight_y}}-1]$

Where

• $Token_x^{Desired}$ : Amount of $Token_x$ being received from the pool (swapped out).

• $Tokeny_{Input}$ : Amount of $Token_y$ being input into the pool (swapped in).

• $Volume_x$​ : Current volume of $Token_x$ in the pool before the transaction.

• $Volume_y$​ : Current volume of $Token_y$ in the pool before the transaction.

• $Weight_x$​ : The current weight of $Token_x$ in the pool.

• $Weight_y$​ : The current weight of $Token_y$ in the pool.

The above formula assumes that the difference between $Volume_x$ and $Token_x^{Desired}$ remains positive and non-zero. As $Token_x^{Desired}$ approaches $Volume_x$, the denominator in the formula approaches zero, causing the calculated $Token_y^{Input}$ to become undefined or extremely large. This scenario represents a situation where the pool is nearly depleted of $Token_x$, making it impractical or impossible to perform the swap.