ybBTC

ybBTC is the generalized description of the LP token representing a share in the Yield Basis pools. Its core feature is impermanent loss mitigation, which protects capital used for liquidity provision from depreciation in the event of volatile asset price movements.

Therefore, this section explains the design of the (ERC20 BTC - crvUSD) pool and the mathematical underpinnings of IL protection.

1. YBBTC POOL DESIGN FOR IMPERMANENT LOSS MITIGATION

ybBTC is the liquidity position (or a pool share) of the (ERC20 BTC - crvUSD) pool. This pool has three core design features:

1. Basic Design: The pool uses Curve's CryptoSwap AMM concentrated liquidity pool.
2. Leverage: The user's LP position (ERC20 BTC + crvUSD) is used as collateral to mint crvUSD. Minted crvUSD is deposited into the pool, increasing the position size. This applies leverage to the user's position, enhances capital efficiency, and makes the position value correspond to the BTC value, not the square root.

This design establishes a mathematical foundation for mitigating IL:

$\displaystyle \text{YB Leverage pool} \propto \text{BTC price}, \quad \text{Pool without leverage} \propto \sqrt{\text{BTC price}}$

3. Rebalancing: The pool's ERC20 BTC and crvUSD balances are automatically adjusted to be in the concentrated liquidity range, optimizing liquidity depth for the pool's users.

The rebalancing process optimizes liquidity by using 50% of the pool's trading fees (ERC20 BTC and crvUSD). The share of the crvUSD borrow rate is also used for rebalancing the position.

2. IN-DEPTH EXPLANATION OF IL MITIGATION

As mentioned, the leverage applied to the user's LP position in the pool is the core design feature that provides mathematical requirements for IL mitigation.

The in-depth explanation of leverage and its IL mitigation function:

1. The Yield Basis leverage algorithm uses the (ERC20 BTC - crvUSD) position as collateral for a crvUSD loan. Its size corresponds to the crvUSD part of the position and is added to the user's position, doubling the value change rate in case of BTC price movements.

2. Due to $\times 2$ leverage, the rate of position value change (growth or decay) equals $\times 2$ that of the unleveraged one. It squares the position value ($v_u^2$) compared to the unleveraged one, mitigating the impermanent loss in this AMM pool.

Statement: application of $\times 2$ leverage to the position results in squaring the position value ($v_u^2$).

Proof:

1. We start from the fact that $\times 2$ leverage means position value grows or declines $\times 2$ faster than the unleveraged one.

The relative value changing rate for the LP position is the value change $\Delta V_l$ divided by the value itself $V_l$. In the continuous limit it reduces to $\frac{dV_l}{V_l}$:

$\frac{dv_l}{v_l} = 2 \cdot \frac{dv_u}{v_u}$

• $v_l$ is the leveraged position's value
• $v_u$ is the unleveraged position's value
• $\frac{dv_u}{v_u}$ and $\frac{dv_l}{v_l}$ are the relative rates of change for the value of the unleveraged and leveraged position

2. Integrating both sides of the equation (constants are discarded):

$\ln v_l = 2\ln v_u$

3. Exponentiating both sides of the equation:

$v_l = v_u^2$

• Squaring the position size ($v_u^2$) eliminates the Impermanent Loss in the Yield Basis pools.

Below we outline the fundamental reasons for impermanent loss in AMM pools and demonstrate how ($v_u^2$) solves it:

1. Let there be a classic cpAMM pool with tokens $X$ and $Y$, all prices denominated in $Y$, and the amounts denominated by the same symbols. By definition of the cpAMM:

$XY = L^2\text{ and }P = \frac{Y}{X}$

2. Expressing the token amounts in terms of the spot price and the AMM invariant:

$Y = \frac{L^2}{X}\implies Y^2 = \frac{YL^2}{X} \implies Y^2 = PL^2 \implies \begin{cases}Y^2 = PL^2 \\ X = \frac{L}{Y} = \frac{L}{\sqrt{P}}\end{cases}$

3. Deriving the value of the liquidity provider's position in the AMM:

$V = Y+PX = L\sqrt{P} + \frac{LP}{\sqrt{P}} = 2L\sqrt{P}\implies V \propto \sqrt{P}$

We demonstrated that Liquity Provider's (LP) position value in the ordinary AMM pool is proportional to $\sqrt{P}$.

This means that liquidity providers don't receive the full appreciation of position value corresponding to the pool's assets' price movements against each other. Even more, they have a money loss compared to holding the same asset portfolio outside the AMM pool. It is an infamous and well-known problem Impermanent Loss Problem, inherent in all AMMs.

Yield Basis solved this problem by adding the built-in leverage function to the pool design. As a result of the leverage, the Liquidity Provider's position value is the square of the originally provided liquidity to the pool ($v_u^2$).

In turn, it leads to $v_{YB\_Position} \propto v^2_{AMM} \propto \sqrt{P} \implies v_{YB\_Position}\propto P$.

Yield Basis design leads to the following ybBTC features:

• The LP position (ybBTC token) in Yield Basis fully appreciates with the pool assets price movements
• IL is mitigated
• LPs generate yield from trading fees, avoiding capital depreciation risks typical for other AMMs.