amm.md

AMM

The AMM strategy seeks to emulate the liquidity available in a concentrated liquidity AMM. Note that max_collateral only bounds the amount of capital used as liquidity at any given time. The only way to bound your total losses is to limit the amount of collateral in your account.

Config

{
    p_min,
    p_max,
    spread,
    delta,
    depth,
    max_collateral
}

Pool Setup

Let $\text{Price}_A$, $\text{Price}_B$ be the midpoint prices of the two tokens, and let $\text{Pool}_A$ and $\text{Pool}_B$ be two concentrated liquidity pools: $\text{Pool}_A$ for the $\text{Token}_A:\text{Collateral}$ pair and $\text{Pool}_B$ for the $\text{Token}_B:\text{Collateral}$ pair.

For each $\text{Pool}$, we consider two liquidity bands, the first, $\text{L}$, the "left" band, in the range $[\text{price} - \text{delta}, \text{price}]$, and the second, $\text{R}$, the "right" band, in the range $[\text{price}, \text{price} + \text{delta}]$. Since $\text{L}$ consists only of prices less than or equal to $\text{price}$, any liquidity provided to $\text{L}$ will consist only of $\text{Collateral}$. Likewise, since $\text{R}$ consists only of prices greater than or equal to $\text{price}$, any liquidity provided to $\text{R}$ will consist only of $\text{Token}$.

Now given quantities $\text{TokenDeposit}_A$ and $\text{CollateralDeposit}_A$, an LP deposits all of $\text{TokenDeposit}_A$ to $\text{L}_A$ and all of $\text{TokenDeposit}_A$ to $\text{R}_A$. Imagining that this is the only liquidity in $\text{Pool}_A$, this completely determines the relationship between size and cost for a single swap in $\text{Pool}_A$.

The LP does the same with $\text{Pool}_B$ given quantities $\text{TokenDeposit}_B$ and $\text{CollateralDeposit}_B$.

Order Placement

Buy Orders

The strategy places buy orders at prices

$$ \text{Prices} = {P_1, P_2, \dots, P_k}, $$

where

$$ \begin{align*} P_1 &= P - \text{spread}, \\ P_k &= P - \text{depth}, \text{ and} \\ P_{i+1} &= P_i - \text{delta} \text{ for } i \in [0, k-1]. \end{align*} $$

To $\text{Prices}$, we assign corresponding sizes

$$ \text{Sizes} = {S_1, S_2, \dots, S_k}, $$

and orders,

$$ \text{Orders} = {O_1, O_2, \dots, O_k}, $$

where, for each $i$, $O_i$ has size $S_i$ and price $P_i$.

To determine the size $S_i$ for a buy order $O_i$, we compute the size, i.e. the $\text{Token}$ amount, $S'_i$ required to move the marginal price of $\text{Pool}$ from $P$ to $P_i$. In other words, if $S_i'$ tokens are swapped into $\text{Pool}$, the resulting marginal price would be $P_i$. Finally define the $S_i$ as follows:

$$ S_i = \begin{cases} S'_1, & \text{if }; i = 1, \text{ and}\ S'i - S'{i-1} & \text{for } i \geq 2. \end{cases} $$

The sizes are chosen such that if these were the only orders on the order book, a trader would sell $S_i$ tokens to fill up every order down to and including $O_i$.

Sell Orders

The same process is used to determine the sizes of sell orders, except that we define $\text{Prices}$ as follows:

$$ \text{Prices} = {P_1, P_2, \dots, P_k}, $$

where

$$ \begin{align*} P_1 &= P + \text{spread}, \\ P_k &= P + \text{depth}, \text{ and} \\ P_{i+1} &= P_i + \text{delta} \text{ for } i \in [0, k-1]. \end{align*} $$