AMM Curve (C)

TermMax AMM Curve

According to the design of concentrated liquidity in Uniswap V3, we can define an AMM curve ($\text{Eq.M-1}$) which acts like a constant product pool with larger reserves within a specific price range $[p_a,p_b]$ ( current price $p_c \in [p_a, p_b]$ ). The real reserves of the two tokens in the pool can be disproportionate when the tokens are traded within the defined range.

Fig. 1 Real Reserves v.s. Virtual Reserves (ref: Uniswap V3 Whitepaper)

$$(x + \frac{L}{\sqrt{p_b}}) \cdot (y + L\sqrt{p_a}) = L^{2} \text{ (Eq.C-1)}\\ \text{where } \\ x :=\text{the real reserve of token0} \\ y :=\text{the real reserve of token1} \\ p_a:=\text{lower bound of token1 to token0 price} \\ p_b:=\text{upper bound of token1 to token0 price} \\ L:=\text{virtual liquidity of the pool} \\ x+\frac{L}{\sqrt{p_b}}:=\text{virtual reserve of token0} \\ y+L\sqrt{p_a}:=\text{virtual reserve of token1}$$

In TermMax, we put our Fixed-Rate Token (FT) and X Token (XT) as token0 and token1 in the AMM pool. FT and XT are minted by the Underlying Token (UT, e.g. USDC) and Initial LTV (ILTV, e.g. 0.9) defined in the pool ($\text{Eq.M-2}$). Liquidity providers can provide 1 USDC and mint 0.9 FT (worth 0.9 USDC at maturity) and 1 XT (worth 0.1 USDC at maturity) and the market price will remain the same.

$$\text{mint FT and XT from UT} \\ UT:FT:XT=1:\varepsilon:1\text{ (Eq.C-2)}$$

As mentioned above, the real reserves of FT and XT provided in the pool are disproportionate. Therefore, we apply the same AMM curve as Uniswap V3 in TermMax.

In addition, to adjust the slippage according to liquidity in the pool and market demand, we define a Liquidity Scaling Factor ($\gamma$) to increase the virtual liquidity as $\gamma^{-1}L$ and derive virtual liquidity $x'$ and $y'$ from $\text{Eq.C-1}$.

$$\text{subject to } \\ x' \cdot y'=(\gamma^{-1}L)^2 \\ \Rightarrow x'=\gamma^{-1}(x+\frac{L}{\sqrt{p_b}}) \land y'= \gamma^{-1}(y+L\sqrt{p_a}) \\ \Rightarrow \gamma^{-1}(x+\frac{L}{\sqrt{p_b}}) \cdot \gamma^{-1}(y+L\sqrt{p_a}) = (\gamma^{-1}L)^2 \text{ (Eq.C-3)}$$

In TermMax, we set $p_b=\infin$ (the upper bound of the XT to FT price) and derive $\text{Eq.C-4}$ from $\text{Eq.C-3}$

$$\gamma^{-1}x \cdot \gamma^{-1}(y + L\sqrt{p_a}) = \gamma^{-2}L^{2} \text{ (Eq.C-4)}$$

With variable substitution, we can derive $\text{Eq.C-5}$ from $\text{Eq.C-4}$

$$x' \cdot y'=k \text{ (Eq.C-5)} \\ \text{where} \\ x':=\gamma^{-1} x \\ y':=\gamma^{-1} \cdot (y+\beta) \\ k:=\gamma^{-2}L^2 \\ \beta:=L\sqrt{p_a}$$

We will use $\text{Eq.C-5}$ to derive the formula for each pool operation.