Adjustable Bonding Curves

The adjustable bonding curves in Carbon provide a flexible yet controlled way to manage liquidity, ensuring stability and minimizing risks of arbitrage in the trading process. Carbon bonding curves are defined by a few key parameters that control their shape and behavior. These curves are important because they help determine how liquidity is managed and traded within the AMM.

Key Parameters of Carbon Bonding Curves

1. $P_0$: A price-like parameter that influences the shape of the curve.

2. y$_0$ A size-like parameter expressed in terms of the risk asset.

3. $Γ$: A unit-less parameter describing the curve's shape.

4. $\chi$: Another size-like parameter in terms of the risk asset, but this can be converted to a form using different units without changing the curve's shape.

In Carbon, liquidity is asymmetric, meaning the two tokens in a pool aren't treated equally. The token being sold determines the pool's activity. Once the pool runs out of tokens to sell, trading stops until it gets more tokens. This token is represented on the y-axis in our models.

Ignoring the constant-product case, which is a special situation, we can describe our bonding curves using three main financial parameters:

1. $P_{start}$: The starting price where the AMM begins selling the active token.

2. $P_{end}$: The ending price where the AMM stops selling the active token.

3. Capacity ($y_{int}$): How much collateral is needed to move from the start price to the end price.

The state of the AMM is described by the amount of tokens available ($y$), which ranges from $0$ to $y_{int}$. These parameters together determine the trading curve and the marginal price, which is crucial to avoid creating arbitrage opportunities.

Marginal Price ($P_{marg}$): This is the price at which the next trade would occur. For $y$ = $y_{int}$, $P_{marg}$ = $P_{start}$, and for $y = 0$, $P_{marg}$= $P_{end}$. The marginal price helps ensure that trades happen at fair market values and prevents arbitrage.

Adjusting the Bonding Curve

When adjusting the bonding curves, there are two main scenarios to consider:

1. Out-of-the-Money: The current market price is far below the starting price. Here, there's a lot of flexibility in adjusting the parameters and state of the curve since the market price is distant.

2. In-the-Money: The market price is close to or within the starting price range. Adjustments need to be more careful to avoid changing the marginal price too much, which could create arbitrage opportunities.

In the first scenario, all parameters and the state can be changed independently. In the second scenario, if the AMM is to remain at the current marginal price, only three parameters can be changed independently. This restricts the adjustments to a specific 3-dimensional space within the larger parameter space.

Practical Considerations

In practice, linked curves (two curves connected to manage collateral) should be non-overlapping to avoid complications. As long as one curve is generating collateral for the other, it stays out-of-the-money, allowing for easier adjustments.

When making adjustments via a user interface, it's important to ensure that changes do not inadvertently create arbitrage opportunities. Therefore, parameters should be changed in pairs to maintain balance.

To simplify the example of adjusting the range of the curve, consider the relationship between the starting price (Pstart) and the ending price (Pend). By understanding this relationship, we can make necessary adjustments to the curve's parameters to keep the trading process efficient and secure.