ConstantSum

Constant Sum Market Maker

This will be all the background needed to understand the ConstantSum DFMM.

Conceptual Overview

The ConstantSum DFMM gives the LP a portfolio that will allow exchange of a pair of tokens at a single price. We can allow this price to be dynamically chosen.

Core

We mark reserves as:

• $x \equiv \mathtt{rX}$
• $y \equiv \mathtt{rY}$

ConstantSum has one variable parameter:

• $P \equiv \mathtt{price}$

The trading function is: $$ \boxed{\varphi(x,y,L;P) =P \frac{x}{L} + \frac{y}{L} -1} $$ where $L$ is the liquidity of the pool.

Price

The reported price of the pool given the reserves is $P$.

Pool initialization

The ConstantSum pool can be initialized with any given price and any given value of reserves. A user may supply $(x_0,y_0,P)$, then we find that:

$$ L_0 = Px_0 + y_0 $$

Swap

We require that the trading function remain invariant when a swap is applied, that is: $$ P\frac{x+\Delta_X}{L + \Delta_L} + \frac{y+\Delta_Y}{L + \Delta_L}-1 = 0 $$

where either $\Delta_X$ or $\Delta_Y$ is given by user input and the $\Delta_L$ comes from fees.

Trade in $\Delta_X$ for $\Delta_Y$

If we want to trade in $\Delta_X$ for $\Delta_Y$, we first accumulate fees by taking $$\Delta_L = (1-\gamma) \Delta_X.$$ Then we can use our invariant equation and solve for $\Delta_Y$ in terms of $\Delta_X$ to get: $$\boxed{\Delta_Y = \gamma P \Delta_X}$$

Trade in $\Delta_Y$ for $\Delta_X$

If we want to trade in $\Delta_X$ for $\Delta_Y$, we first accumulate fees by taking $$\Delta_L = \frac{1-\gamma}{P}\Delta_Y.$$ Then we can use our invariant equation and solve for $\Delta_X$ in terms of $\Delta_Y$ to get: $$\boxed{\Delta_X = \frac{\gamma}{P} \Delta_Y}$$

Allocations and Deallocations

Allocations and deallocations should not change the price of a pool and since this pool only quotes a single price, any amount of reserves can be allocated at any time. We need only compute the new $L$. Specifically:

$$ \Delta_L = P\Delta_X + \Delta_Y $$

Value Function via $L$ and $S$

Given that we treat $Y$ as the numeraire, we know that the portfolio value of a pool when $X$ is at price $S$ is: $$V = Sx(S) + y(S)$$

In this case, the value function is that of a limit order and follows: $$V(L,S) = \begin{cases} LS & \text{if } S \leq P \ LP & \text{if } S \geq P \end{cases}$$